Polynomial remainder theorem proof

WEBVTT
Kind: captions
Language: en

00:00:00.417 --> 00:00:01.590
- [Voiceover] Let's now do a proof

00:00:01.590 --> 00:00:04.248
of the polynomial remainder theorem.

00:00:04.248 --> 00:00:06.257
Just to make the proof
a little bit tangible,

00:00:06.257 --> 00:00:07.731
I'm going to start with the example

00:00:07.731 --> 00:00:09.948
that we saw in the video that introduced

00:00:09.948 --> 00:00:11.516
the polynomial remainder theorem.

00:00:11.516 --> 00:00:13.129
We saw that if you took three x squared

00:00:13.129 --> 00:00:15.068
minus four x plus seven and you divided

00:00:15.068 --> 00:00:18.307
by x minus one, you got three x minus one

00:00:18.307 --> 00:00:20.153
with the remainder of six.

00:00:20.153 --> 00:00:21.732
When we do polynomial long division,

00:00:21.732 --> 00:00:23.530
how do we know when we've
got to our remainder?

00:00:23.530 --> 00:00:25.183
Well when we get to an expression that has

00:00:25.183 --> 00:00:28.689
a lower degree than the
thing that is the divisor,

00:00:28.689 --> 00:00:32.322
the thing that we're dividing
into the other thing.

00:00:32.322 --> 00:00:34.862
So in this example we
could have re-written

00:00:34.862 --> 00:00:39.302
what we just did right
over here as our f of x.

00:00:39.302 --> 00:00:40.719
Let me just write it right over here.

00:00:40.719 --> 00:00:44.498
So we could have said three x squared

00:00:44.498 --> 00:00:48.480
minus four x plus seven is equal to

00:00:48.480 --> 00:00:51.359
x minus one times the
quotient right over here,

00:00:51.359 --> 00:00:53.797
or I could say the
quotient times x minus one.

00:00:53.797 --> 00:00:57.384
So it's going to be
equal to this business.

00:00:57.384 --> 00:01:01.877
It's going to be equal
to three x minus one

00:01:01.877 --> 00:01:06.138
times the divisor, times x minus one.

00:01:10.497 --> 00:01:11.566
When you multiply these two things,

00:01:11.566 --> 00:01:12.611
you're not going to get exactly this.

00:01:12.611 --> 00:01:14.665
You still have to add the remainder.

00:01:14.933 --> 00:01:18.391
So plus the remainder.

00:01:18.694 --> 00:01:20.535
Actually, let me write
the actual remainder down.

00:01:20.535 --> 00:01:22.309
So plus six.

00:01:22.346 --> 00:01:24.587
The analogy here is exactly the analogy

00:01:24.587 --> 00:01:26.444
to when you did traditional division.

00:01:26.444 --> 00:01:29.370
If I were to say, actually, let me just

00:01:29.370 --> 00:01:30.924
show you the analogy.

00:01:31.088 --> 00:01:36.088
If I were to say 25 divided by four,

00:01:38.529 --> 00:01:42.627
you would say, okay,
well four goes into 25

00:01:42.627 --> 00:01:46.121
six times, six times four is 24.

00:01:46.284 --> 00:01:48.397
You would subtract, and then you would get

00:01:48.397 --> 00:01:51.065
a remainder, one.

00:01:51.299 --> 00:01:53.505
Or another way of saying
this is you could say

00:01:53.505 --> 00:01:58.505
that 25 is equal to six
times four plus one.

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So we just did the exact same thing here,

00:02:13.948 --> 00:02:16.338
but we just did it with expressions.

00:02:16.664 --> 00:02:18.801
So once again, I haven't
started the proof yet,

00:02:18.801 --> 00:02:20.125
I just wanted to make you feel comfortable

00:02:20.125 --> 00:02:22.168
with what I just wrote right over here.

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If I divided this expression
into this polynomial,

00:02:26.242 --> 00:02:27.705
and I were to get this quotient,

00:02:27.705 --> 00:02:29.564
that's the same thing as
saying this polynomial

00:02:29.564 --> 00:02:32.186
could be equal to three x minus one

00:02:32.186 --> 00:02:35.156
times x minus one plus six.

00:02:35.599 --> 00:02:37.201
Now this is true in general.

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Let's abstract a little bit.

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This is our f of x.

00:02:41.276 --> 00:02:43.120
So that is f of x.

00:02:45.235 --> 00:02:47.104
So f of x is going to be equal to

00:02:47.104 --> 00:02:48.578
whatever the quotient is.

00:02:48.578 --> 00:02:50.360
Let me call that q of x.

00:02:51.030 --> 00:02:52.922
So do this in a different color.

00:02:53.722 --> 00:02:58.333
So I'm going to call that q of x.

00:02:59.855 --> 00:03:02.021
This right over here is q of x.

00:03:02.220 --> 00:03:04.437
So f of x is going to be
equal to the quotient,

00:03:04.437 --> 00:03:08.036
q of x, times, this is our x minus a,

00:03:08.036 --> 00:03:09.499
in this case a is one, but I'm just trying

00:03:09.499 --> 00:03:11.601
to generalize it a little bit.

00:03:11.601 --> 00:03:16.098
So x minus a, and then plus the remainder.

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We know that the remainder is going

00:03:19.532 --> 00:03:21.746
to be a constant because the remainder

00:03:21.746 --> 00:03:26.024
is going to have a lower
degree then x minus a.

00:03:26.024 --> 00:03:28.288
X minus a is a first degree.

00:03:28.288 --> 00:03:30.006
So in order to have a lower degree,

00:03:30.006 --> 00:03:31.307
this has to be zeroth degree.

00:03:31.307 --> 00:03:33.280
This has to be a constant.

00:03:33.280 --> 00:03:35.160
So this is true in general.

00:03:35.160 --> 00:03:37.145
This is true for any polynomial

00:03:37.145 --> 00:03:42.145
f of x divided by any x minus a.

00:03:42.486 --> 00:03:43.992
So this is just true.

00:03:44.586 --> 00:03:49.586
So this is true for any
f of x and x minus a.

00:03:55.070 --> 00:04:00.070
Now what is going to happen
if we evaluate f of a?

00:04:04.032 --> 00:04:07.805
Well if f of x can be written like this,

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well we could write f of,

00:04:10.347 --> 00:04:14.027
let me do a in a new
color so it sticks out.

00:04:14.027 --> 00:04:19.027
We could write f of a would be equal to

00:04:22.444 --> 00:04:27.444
q of a times, I think you might see

00:04:31.660 --> 00:04:36.660
where this is going,
times a minus a plus r.

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Well what's that going to be equal to?

00:04:45.110 --> 00:04:47.071
What's all this business
going to be equal to?

00:04:47.071 --> 00:04:51.019
Well a minus a is zero, and q of a,

00:04:51.019 --> 00:04:52.296
I don't care what q of a is,

00:04:52.296 --> 00:04:53.829
if you're going to multiply it by zero

00:04:53.829 --> 00:04:56.233
all of this is going to be zero.

00:04:56.233 --> 00:05:01.233
So f of a is going to be equal to r.

00:05:06.634 --> 00:05:07.829
And so you're done.

00:05:07.829 --> 00:05:09.339
This is the proof of the

00:05:09.339 --> 00:05:11.196
polynomial remainder theorem.

00:05:11.196 --> 00:05:15.515
Any function, if when you
divide it by x minus a

00:05:15.515 --> 00:05:18.580
you get the quotient q
of x and the remainder r,

00:05:18.580 --> 00:05:20.390
it can then be written in this way.

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If it's written in this
way and you evaluated

00:05:22.701 --> 00:05:25.405
at f of a and you put the a over here,

00:05:25.405 --> 00:05:26.926
you're going to see that
f of a is going to be

00:05:26.926 --> 00:05:29.862
whatever that remainder was.

00:05:29.862 --> 00:05:33.625
That is the polynomial remainder theorem.

00:05:33.625 --> 00:05:34.646
And we're done.

00:05:34.646 --> 00:05:36.998
One of the simpler proofs that exists

00:05:36.998 --> 00:05:40.735
for something that at first
seems somewhat magical.


Polynomial remainder theorem

WEBVTT
Kind: captions
Language: en

00:00:00.371 --> 00:00:02.145
- [Voiceover] So let's introduce ourselves

00:00:02.145 --> 00:00:06.277
to the Polynomial Remainder Theorem.

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And as we'll see a little,

00:00:07.436 --> 00:00:09.315
you'll feel a little magical at first.

00:00:09.315 --> 00:00:11.415
But in future videos, we will
prove it and we will see,

00:00:11.415 --> 00:00:13.063
well, like many things in Mathematics.

00:00:13.063 --> 00:00:14.445
When you actually think it through,

00:00:14.445 --> 00:00:16.923
maybe it's not so much magic.

00:00:16.923 --> 00:00:19.469
So what is the Polynomial
Remainder Theorem?

00:00:19.469 --> 00:00:21.569
Well it tells us that if we start

00:00:21.569 --> 00:00:24.313
with some polynomial, f of x.

00:00:24.313 --> 00:00:27.793
So this right over here is a polynomial.

00:00:27.793 --> 00:00:29.557
Polynomial.

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And we divide it

00:00:34.923 --> 00:00:39.332
by x minus a.

00:00:39.332 --> 00:00:43.890
Then the remainder

00:00:43.890 --> 00:00:46.103
from that essentially polynomial

00:00:46.103 --> 00:00:49.916
long division is going to be f of a.

00:00:49.916 --> 00:00:53.040
It is going to be

00:00:53.040 --> 00:00:56.887
f of a.

00:00:56.887 --> 00:00:59.330
I know this might seem a
little bit abstract right now.

00:00:59.330 --> 00:01:02.727
I'm talking about f of
x's and x minus a's.

00:01:02.727 --> 00:01:05.409
Let's make it a little bit more concrete.

00:01:05.409 --> 00:01:10.409
So let's say that f of x is equal to,

00:01:10.455 --> 00:01:11.737
I'm just gonna make up a,

00:01:11.737 --> 00:01:13.302
let's say a second degree polynomial.

00:01:13.302 --> 00:01:15.176
This would be true for
any polynomial though.

00:01:15.176 --> 00:01:18.078
So three x squared minus

00:01:18.078 --> 00:01:21.139
four x plus seven.

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And let's say that a is,
I don't know, a is one.

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So we're gonna divide that by,

00:01:30.607 --> 00:01:33.886
we're going to divide by

00:01:33.886 --> 00:01:38.886
x minus one.

00:01:39.006 --> 00:01:44.006
So a, in this case, is equal to one.

00:01:44.019 --> 00:01:45.890
So let's just do the
polynomial long division.

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I encourage you to pause the video.

00:01:47.665 --> 00:01:49.635
If you're unfamiliar with
polynomial long division,

00:01:49.635 --> 00:01:51.815
I encourage you to watch that
before watching this video

00:01:51.815 --> 00:01:53.327
because I will assume you know

00:01:53.327 --> 00:01:55.223
how to do a polynomial long division.

00:01:55.223 --> 00:01:57.983
So divide three x squared
minus four x plus seven.

00:01:57.983 --> 00:01:59.477
Divide it by x minus one.

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See what you get as the remainder

00:02:01.036 --> 00:02:04.877
and see if that remainder
really is f of one.

00:02:04.877 --> 00:02:06.422
So assuming you had a go at it.

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So let's work through it together.

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So let's divide x minus one

00:02:13.379 --> 00:02:18.379
into three x squared

00:02:18.752 --> 00:02:22.364
minus four x plus seven.

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All right, little bit of
polynomial long division is

00:02:24.907 --> 00:02:26.745
never a bad way to start your morning.

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It's morning for me.

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I don't know what it is for you.

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All right, so I look at the x term here,

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the highest degree term.

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And then I'll start with the
highest degree term here.

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So how many times does x
going to three x squared?

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What was three x times?

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Three x times x is three x squared.

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So I'll write three x over here.

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I'll write it in the, I could say

00:02:47.920 --> 00:02:49.700
the first degree place.

00:02:49.700 --> 00:02:53.750
Three x times x is three x squared.

00:02:53.750 --> 00:02:57.822
Three x times negative
one is negative three x.

00:02:57.822 --> 00:03:01.486
And now we want to subtract this thing.

00:03:01.486 --> 00:03:04.454
It's just the way that you
do traditional long division.

00:03:04.454 --> 00:03:06.505
And so, what do we get?

00:03:06.505 --> 00:03:09.488
Well, three x squared
minus three x squared.

00:03:09.488 --> 00:03:11.552
That's just going to be a zero.

00:03:11.552 --> 00:03:14.237
So this just add up to zero.

00:03:14.237 --> 00:03:16.584
And this negative four x,

00:03:16.584 --> 00:03:18.332
this is going to be plus three x, right?

00:03:18.332 --> 00:03:19.720
And negative of a negative.

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Negative four x plus three x

00:03:22.012 --> 00:03:25.367
is going to be negative x.

00:03:25.367 --> 00:03:27.504
I'm gonna do this in a new color.

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So it's going to be negative x.

00:03:31.513 --> 00:03:35.705
And then we can bring down seven.

00:03:35.705 --> 00:03:38.346
Complete analogy to how you
first learned long division

00:03:38.346 --> 00:03:40.713
in maybe, I don't know,
third or fourth grade.

00:03:40.713 --> 00:03:42.565
So all I did is I multiplied
three x times this.

00:03:42.565 --> 00:03:44.813
You get three x squared minus three x

00:03:44.813 --> 00:03:46.801
and then I subtract to
that from three x squared

00:03:46.801 --> 00:03:49.255
minus four x to get this right over here

00:03:49.255 --> 00:03:52.518
or you could say I subtract
it from this whole polynomial

00:03:52.518 --> 00:03:55.856
and then I got negative x plus seven.

00:03:55.856 --> 00:03:58.149
So now, how many times does x

00:03:58.149 --> 00:04:00.598
minus one go to negative x plus seven?

00:04:00.598 --> 00:04:02.098
Well x goes into negative x,

00:04:02.098 --> 00:04:06.488
negative one times x

00:04:06.488 --> 00:04:08.816
is negative x.

00:04:08.816 --> 00:04:12.662
Negative one times negative
one is positive one.

00:04:12.662 --> 00:04:15.131
But then we're gonna
wanna subtract this thing.

00:04:15.131 --> 00:04:16.357
We're gonna wanna subtract this thing

00:04:16.357 --> 00:04:18.660
and this is going to
give us our remainder.

00:04:18.660 --> 00:04:21.616
So negative x minus negative x.

00:04:21.616 --> 00:04:24.713
Just the same thing as negative x plus x.

00:04:24.713 --> 00:04:26.847
These are just going to add up to zero.

00:04:26.847 --> 00:04:27.939
And then you have seven.

00:04:27.939 --> 00:04:29.104
This is going to be seven plus one.

00:04:29.104 --> 00:04:30.188
Remember you have this negative out

00:04:30.188 --> 00:04:31.329
so if you distribute the negative,

00:04:31.329 --> 00:04:33.144
this is going to be a negative one.

00:04:33.144 --> 00:04:35.880
Seven minus one is six.

00:04:35.880 --> 00:04:39.709
So your remainder here is six.

00:04:39.709 --> 00:04:40.982
One way to think about it,

00:04:40.982 --> 00:04:45.442
you could say that, well (mumbles).

00:04:45.442 --> 00:04:46.797
I'll save that for a future video.

00:04:46.797 --> 00:04:50.612
This right over here is the remainder.

00:04:50.612 --> 00:04:52.128
And you know when you
got to the remainder,

00:04:52.128 --> 00:04:54.609
this is just all review of
polynomial long division,

00:04:54.609 --> 00:04:57.128
is when you get something
that has a lower degree.

00:04:57.128 --> 00:04:58.676
This is, I guess you could call this

00:04:58.676 --> 00:05:01.058
a zero degree polynomial.

00:05:01.058 --> 00:05:04.225
This has a lower degree
than what you are actually

00:05:04.225 --> 00:05:09.225
dividing into or than the x
minus one than your divisor.

00:05:09.680 --> 00:05:11.852
So this a lower degree
so this is the remainder.

00:05:11.852 --> 00:05:16.014
You can't take this
into this anymore times.

00:05:16.014 --> 00:05:20.471
Now, by the Polynomial Remainder Theorem,

00:05:20.471 --> 00:05:23.538
if it's true and I just
picked a random example here.

00:05:23.538 --> 00:05:26.108
This is by no means a proof but just kinda

00:05:26.108 --> 00:05:29.308
a way to make it tangible of Polynomial

00:05:29.308 --> 00:05:32.009
(laughs) Remainder Theorem is telling us.

00:05:32.009 --> 00:05:34.566
If the Polynomial
Remainder Theorem is true,

00:05:34.566 --> 00:05:38.968
it's telling us that f
of a, in this case, one,

00:05:38.968 --> 00:05:42.727
f of one should be equal to six.

00:05:42.727 --> 00:05:44.600
It should be equal to this remainder.

00:05:44.600 --> 00:05:45.555
Now let's verify that.

00:05:45.555 --> 00:05:48.838
This is going to be equal
to three times one squared,

00:05:48.838 --> 00:05:51.860
which is going to be three,
minus four times one,

00:05:51.860 --> 00:05:55.717
so that's just going to
be minus four, plus seven.

00:05:55.717 --> 00:06:00.017
Three minus four is negative
one plus seven is indeed,

00:06:00.017 --> 00:06:01.635
we deserve a minor drumroll,

00:06:01.635 --> 00:06:04.606
is indeed equal to six.

00:06:04.606 --> 00:06:07.604
So this is just kinda, at
least for this particular case,

00:06:07.604 --> 00:06:09.082
looks like okay, it
seems like the Polynomial

00:06:09.082 --> 00:06:10.415
Remainder Theorem worked.

00:06:10.415 --> 00:06:12.365
But the utility of it is if someone said,

00:06:12.365 --> 00:06:15.111
"Hey, what's the remainder
if I were to divide

00:06:15.111 --> 00:06:16.986
"three x squared minus four x plus seven

00:06:16.986 --> 00:06:19.714
"by x minus one if all I
care about is the remainder?"

00:06:19.714 --> 00:06:21.865
They don't care about the actual quotient.

00:06:21.865 --> 00:06:23.903
All they care about is
the remainder, you could,

00:06:23.903 --> 00:06:27.299
"Hey, look, I can just take
that, in this case, a is one.

00:06:27.299 --> 00:06:28.361
"I can throw that in.

00:06:28.361 --> 00:06:30.764
"I can evaluate f of one
and I'm gonna get six.

00:06:30.764 --> 00:06:32.068
"I don't have to do all of this business.

00:06:32.068 --> 00:06:34.104
"All I had, would have to do is this

00:06:34.104 --> 00:06:37.130
"to figure out the remainder
of three x squared."

00:06:37.130 --> 00:06:38.820
Well you take three x
squared minus four plus seven

00:06:38.820 --> 00:06:41.854
and divide by x minus one.


Empirical, molecular, and structural formulas

WEBVTT
Kind: captions
Language: en

00:00:00.512 --> 00:00:02.172
- What I want to do in
this video is think about

00:00:02.172 --> 00:00:06.595
the different ways to
represent a molecule.

00:00:06.595 --> 00:00:10.299
So the most obvious way is its name.

00:00:10.299 --> 00:00:13.050
So, for example, you could be referring to

00:00:13.050 --> 00:00:16.439
a molecule of benzene.

00:00:16.439 --> 00:00:17.717
Benzene.

00:00:17.717 --> 00:00:20.259
But just the word "benzene"
tells you very little

00:00:20.259 --> 00:00:22.755
about what actually
makes up this molecule.

00:00:22.755 --> 00:00:24.717
And there's other naming
conventions that do

00:00:24.717 --> 00:00:26.304
give more information, but you might say,

00:00:26.304 --> 00:00:28.321
well, I actually want to know more about

00:00:28.321 --> 00:00:31.676
the actual particular
elements that make it up.

00:00:31.676 --> 00:00:34.160
Well, that might be, in that case,

00:00:34.160 --> 00:00:37.898
it might be useful to move
up to the empirical formula.

00:00:37.898 --> 00:00:41.900
Empirical, empirical.

00:00:41.900 --> 00:00:44.547
Empirical.

00:00:44.547 --> 00:00:46.674
Empirical formula.

00:00:46.674 --> 00:00:48.509
And you might be thinking, what does

00:00:48.509 --> 00:00:50.433
empirical mean?

00:00:50.433 --> 00:00:53.532
In general, the word "empirical"
is referring to something

00:00:53.532 --> 00:00:57.805
that comes from observation
or comes through experiments.

00:00:57.805 --> 00:01:01.406
If you could say hey, you
know, I from empirical evidence

00:01:01.406 --> 00:01:03.662
I now believe this, this
means that you saw data.

00:01:03.662 --> 00:01:06.508
This means that you have
some observations that

00:01:06.508 --> 00:01:08.470
make you think this new thing.

00:01:08.470 --> 00:01:10.664
The reason why we call what
I'm about to write down

00:01:10.664 --> 00:01:14.554
the empirical formula, is
because early chemists,

00:01:14.554 --> 00:01:16.441
they can't look, they
weren't able to look at

00:01:16.442 --> 00:01:18.716
just one molecule, but
they could at least come up

00:01:18.716 --> 00:01:22.872
with, they could observe
the ratios of the different

00:01:22.872 --> 00:01:25.625
elements that they had in a molecule.

00:01:25.625 --> 00:01:27.715
So an empirical formula gives you a ratio

00:01:27.715 --> 00:01:30.604
of the elements in the molecule.

00:01:30.604 --> 00:01:34.954
So an empirical formula for benzene is ...

00:01:34.954 --> 00:01:37.180
It is ...

00:01:37.180 --> 00:01:39.873
One carbon for

00:01:39.873 --> 00:01:43.550
every, for every hydrogen.

00:01:43.550 --> 00:01:45.260
Now you might say, OK, that's nice,

00:01:45.260 --> 00:01:47.059
I now know that if I'm
dealing with benzene

00:01:47.059 --> 00:01:48.692
I have one carbon for every hydrogen

00:01:48.692 --> 00:01:50.352
or one hydrogen for every carbon,

00:01:50.352 --> 00:01:53.800
but what does, how many of
each of these do you actually

00:01:53.800 --> 00:01:58.014
have in a benzene molecule?

00:01:58.014 --> 00:02:00.522
To answer that question,
that's when you would want

00:02:00.522 --> 00:02:03.517
to go to the molecular formula.

00:02:03.517 --> 00:02:05.816
Molecular.

00:02:05.816 --> 00:02:09.042
Molecular formula.

00:02:09.042 --> 00:02:11.121
And the molecular formula
for benzene, which is

00:02:11.121 --> 00:02:13.092
now going to give us more information than

00:02:13.092 --> 00:02:16.902
the empirical formula,
tells us that each benzene

00:02:16.902 --> 00:02:21.310
molecule has six hydrogens, and,

00:02:21.310 --> 00:02:24.471
sorry, six carbons and six,

00:02:24.471 --> 00:02:27.126
(laughs) I'm really having trouble today,

00:02:27.126 --> 00:02:31.526
six hydrogens, (laughs) six carbons, and,

00:02:31.526 --> 00:02:32.954
six hydrogens.

00:02:32.954 --> 00:02:34.773
Now, the ratio is still
one to one, you get that

00:02:34.773 --> 00:02:36.932
right over here, it's very easy to go from

00:02:36.932 --> 00:02:39.870
a molecular formula to
an empirical formula.

00:02:39.870 --> 00:02:41.849
You essentially are losing information.

00:02:41.849 --> 00:02:43.178
You're just saying the ratio, OK, look,

00:02:43.178 --> 00:02:44.594
it's a ratio of six to six,

00:02:44.594 --> 00:02:46.671
which is the same thing as one to one.

00:02:46.671 --> 00:02:51.436
If we wanted to, we
could write this as C one

00:02:51.436 --> 00:02:53.972
H one just like that to
show us that the ratio

00:02:53.972 --> 00:02:56.248
for every carbon we have a hydrogen.

00:02:56.248 --> 00:02:58.843
And we see that that's actually
the case in one molecule,

00:02:58.843 --> 00:03:01.089
for every six carbons
you have six hydrogens,

00:03:01.089 --> 00:03:03.411
which is still a one to one ratio.

00:03:03.411 --> 00:03:05.654
That may not satisfy you, you might say,

00:03:05.654 --> 00:03:07.849
well, OK, but how are these six carbons

00:03:07.849 --> 00:03:10.325
and six hydrogens actually structured?

00:03:10.325 --> 00:03:12.558
I want more information.

00:03:12.558 --> 00:03:15.263
And for that, you would wanna go to

00:03:15.263 --> 00:03:17.979
a structural formula.

00:03:17.979 --> 00:03:20.755
Structural formula,

00:03:20.755 --> 00:03:23.296
which will actually
give you the structure,

00:03:23.296 --> 00:03:26.455
or start to give you the
structure of a benzene molecule.

00:03:26.455 --> 00:03:30.263
A benzene molecule would be drawn like...

00:03:30.263 --> 00:03:33.676
So you would have six
carbons in a hexagon.

00:03:33.676 --> 00:03:37.739
So one, going to write this way, one, two,

00:03:37.739 --> 00:03:42.463
three, four, five, six

00:03:42.463 --> 00:03:44.786
carbons in a hexagon just like that.

00:03:44.786 --> 00:03:47.432
And then you have a
double bond, every other

00:03:47.432 --> 00:03:51.473
of these bonds on the
hexagon is a double bond.

00:03:51.473 --> 00:03:54.238
Each of these carbons are
also attached to a hydrogen,

00:03:54.238 --> 00:03:56.178
also bonded to a hydrogen.

00:03:56.178 --> 00:03:58.940
Each of these lines that I'm drawing,

00:03:58.940 --> 00:04:01.041
this is a bond, it's a covalent bond,

00:04:01.041 --> 00:04:03.061
we go into much more depth
in other videos on that,

00:04:03.061 --> 00:04:05.117
but it's a sharing of
electrons, and that's what keeps

00:04:05.117 --> 00:04:07.752
these carbons near each
other and what keeps

00:04:07.752 --> 00:04:09.772
the hydrogens kind of tied to each, or,

00:04:09.772 --> 00:04:11.512
the hydrogens tied to the
carbons and the carbons

00:04:11.512 --> 00:04:12.952
tied to the hydrogens.

00:04:12.952 --> 00:04:15.251
So let me draw it just like this.

00:04:15.251 --> 00:04:19.053
And this is only one
variant of a structural,

00:04:19.053 --> 00:04:22.179
it's hard to see this one I just drew,

00:04:22.179 --> 00:04:24.269
so let me see if I can do a little bit...

00:04:24.269 --> 00:04:26.077
Oh, that's about as good,
hopefully you see there's

00:04:26.077 --> 00:04:28.910
a hydrogen there, and there's
a hydrogen right over there.

00:04:28.910 --> 00:04:30.979
This is one variant of
a structural formula,

00:04:30.979 --> 00:04:32.742
some structural formulas
will actually give you

00:04:32.742 --> 00:04:35.660
some 3D information, will
tell you whether a molecule

00:04:35.660 --> 00:04:37.992
is kind of popping in or out of the page.

00:04:37.992 --> 00:04:40.999
Others might not be as explicit,

00:04:40.999 --> 00:04:43.442
once you go into organic chemistry

00:04:43.442 --> 00:04:47.523
chains of carbons are just
done, they're just ...

00:04:47.523 --> 00:04:50.251
You might see something
like this for benzene,

00:04:50.251 --> 00:04:54.941
where the carbons are implicit
as the vertex of each,

00:04:54.958 --> 00:04:57.415
there's an implicit carbon
at each of these vertices,

00:04:57.415 --> 00:04:59.714
and then you say, OK, carbon's gotta have,

00:04:59.714 --> 00:05:01.757
not gotta, but it's
typically going to have

00:05:01.757 --> 00:05:05.473
four bonds in its stable state,
I only see one, two, three.

00:05:05.473 --> 00:05:07.887
Well, if it's not drawn,
then it must be a hydrogen.

00:05:07.887 --> 00:05:09.460
That's actually the convention that people

00:05:09.460 --> 00:05:10.864
use in organic chemistry.

00:05:10.864 --> 00:05:13.978
So there's multiple ways
to do a structural formula,

00:05:13.978 --> 00:05:16.741
but this is a very typical
one right over here.

00:05:16.741 --> 00:05:18.421
As you see, I'm just getting more and more

00:05:18.421 --> 00:05:20.659
and more information
as I go from empirical

00:05:20.659 --> 00:05:23.653
to molecular to structural formula.

00:05:23.653 --> 00:05:25.940
Now, I want to make clear, that empirical

00:05:25.940 --> 00:05:29.133
formulas and molecular formulas
aren't always different

00:05:29.133 --> 00:05:33.522
if the ratios are actually, also show the

00:05:33.522 --> 00:05:36.138
actual number of each of those elements

00:05:36.138 --> 00:05:37.437
that you have in a molecule.

00:05:37.437 --> 00:05:41.509
A good example of that would be water.

00:05:41.509 --> 00:05:43.780
Let me do water.

00:05:43.780 --> 00:05:45.416
Let me do this in a
different color that I,

00:05:45.416 --> 00:05:48.794
well, I've pretty much
already used every color.

00:05:48.794 --> 00:05:50.280
Water.

00:05:50.280 --> 00:05:54.194
So water we all know,
for every two hydrogens,

00:05:54.194 --> 00:05:56.794
for every two hydrogens, and since I

00:05:56.794 --> 00:05:58.806
already decided to use
blue for hydrogen let me

00:05:58.806 --> 00:06:00.687
use blue again for hydrogen,

00:06:00.687 --> 00:06:05.075
for every two hydrogens
you have an oxygen.

00:06:05.075 --> 00:06:06.863
You have an oxygen.

00:06:06.863 --> 00:06:09.958
It just so happens to be,
what I just wrote down

00:06:09.958 --> 00:06:11.942
I kind of thought of in
terms of empirical formula,

00:06:11.942 --> 00:06:14.335
in terms of ratios, but
that's actually the case.

00:06:14.335 --> 00:06:18.038
A molecule of hydrogen,
sorry, a molecule of water

00:06:18.038 --> 00:06:21.833
has exactly two hydrogens and,

00:06:21.833 --> 00:06:23.934
and one oxygen.

00:06:23.934 --> 00:06:25.745
If you want to see the structural formula,

00:06:25.745 --> 00:06:27.221
you're probably familiar with it or you

00:06:27.221 --> 00:06:28.660
might be familiar with it.

00:06:28.660 --> 00:06:33.501
Each of those oxygens in a water molecule

00:06:33.501 --> 00:06:36.171
are bonded to two hydrogens,

00:06:36.171 --> 00:06:38.910
are bonded to two hydrogens.

00:06:38.910 --> 00:06:42.011
So hopefully this at least begins to

00:06:42.011 --> 00:06:44.906
appreciate different ways of referring to

00:06:44.906 --> 00:06:48.906
or representing a molecule.


Combination formula

WEBVTT
Kind: captions
Language: en

00:00:00.244 --> 00:00:02.050
- The first time you're
exposed to permutations

00:00:02.050 --> 00:00:03.809
and combinations it takes a little bit

00:00:03.809 --> 00:00:06.110
to get your brain around it,
so I think it never hurts

00:00:06.110 --> 00:00:07.163
to do as many examples.

00:00:07.163 --> 00:00:09.778
But each incremental
example I'm gonna review

00:00:09.778 --> 00:00:11.055
what we've done before, but hopefully go

00:00:11.055 --> 00:00:12.634
a little bit further.

00:00:12.634 --> 00:00:13.876
So let's just take another example.

00:00:13.876 --> 00:00:15.744
This is in the same vein.

00:00:15.744 --> 00:00:17.742
In videos after this I'll
start using other examples,

00:00:17.742 --> 00:00:19.472
other than just people sitting in chairs,

00:00:19.472 --> 00:00:20.911
but let's just stick with it for now.

00:00:20.911 --> 00:00:22.641
Let's say we have six people again.

00:00:22.641 --> 00:00:25.408
Person A, B, C,

00:00:25.408 --> 00:00:28.221
D, E, and F,

00:00:28.221 --> 00:00:29.467
so we have six people.

00:00:29.467 --> 00:00:31.362
Now let's put them into four chairs.

00:00:31.362 --> 00:00:32.811
We can go through this fairly quickly,

00:00:32.811 --> 00:00:37.048
one, two, three, four chairs.

00:00:37.048 --> 00:00:39.022
We've seen this show multiple times.

00:00:39.022 --> 00:00:41.367
How many ways, how many
permutations are there

00:00:41.367 --> 00:00:43.549
of putting these six
people into four chairs?

00:00:43.549 --> 00:00:45.341
Well the first chair, if
we seat them in order,

00:00:45.341 --> 00:00:47.043
we might as well, we
can say well there'd be

00:00:47.043 --> 00:00:48.587
six possibilities here.

00:00:48.587 --> 00:00:50.294
For each of those six possibilities

00:00:50.294 --> 00:00:52.569
there would be five possibilities
of who we'd put here,

00:00:52.569 --> 00:00:54.531
because one person's already sitting down.

00:00:54.531 --> 00:00:56.307
For each of these 30 possibilities

00:00:56.307 --> 00:00:57.933
of seating these first two people,

00:00:57.933 --> 00:00:59.428
there'd be four possibilities

00:00:59.428 --> 00:01:02.495
of who we put in chair number three.

00:01:02.495 --> 00:01:05.629
Then for each of these, what
is this, 120 possibilities

00:01:05.629 --> 00:01:07.614
there would be three possibilities

00:01:07.614 --> 00:01:09.855
of who we put in chair four.

00:01:09.855 --> 00:01:14.109
This six times five times four times three

00:01:14.109 --> 00:01:16.012
is the number of permutations.

00:01:16.012 --> 00:01:19.212
We've seen in one of the
early videos on permutations

00:01:19.212 --> 00:01:22.390
that, when we talk about
the permutation formula,

00:01:22.390 --> 00:01:24.924
one way to write this, if we
wanted to write in terms of

00:01:24.924 --> 00:01:28.151
factorial, we could write this as

00:01:28.151 --> 00:01:32.214
six factorial, six factorial
which is going to be equal to

00:01:32.214 --> 00:01:35.557
six times five times four times

00:01:35.557 --> 00:01:38.933
three times two times one,

00:01:38.933 --> 00:01:41.336
but we wanna get rid of the two times one.

00:01:41.336 --> 00:01:43.076
We're gonna divide that,

00:01:43.076 --> 00:01:44.184
we're gonna divide that.

00:01:44.184 --> 00:01:45.616
Now what's two times one?

00:01:45.616 --> 00:01:47.422
Two times one is two factorial.

00:01:47.422 --> 00:01:48.664
Where do we get that?

00:01:48.664 --> 00:01:50.459
We wanted the first four,

00:01:50.459 --> 00:01:54.223
the first four factors of six factorial.

00:01:54.639 --> 00:01:55.944
That's where the four came from.

00:01:55.944 --> 00:01:59.379
We wanted the first four
factors, so the way we got two

00:01:59.379 --> 00:02:03.233
is we said six minus four.

00:02:03.233 --> 00:02:05.590
Six minus four,

00:02:05.590 --> 00:02:08.353
that's gonna give us the number

00:02:08.353 --> 00:02:10.361
that we wanna get rid of.

00:02:10.361 --> 00:02:11.522
We wanted to get rid of two

00:02:11.522 --> 00:02:13.484
or the factors we wanna get rid of.

00:02:13.484 --> 00:02:15.376
That's going to give us two factorial.

00:02:15.376 --> 00:02:18.174
If we do six minus four factorial.

00:02:18.174 --> 00:02:20.774
That's going to give us two factorial,

00:02:20.774 --> 00:02:23.677
which is two times one.

00:02:23.677 --> 00:02:27.357
Then these cancel out, and we are all set.

00:02:27.357 --> 00:02:31.443
This is one way, I put in
the particular numbers here,

00:02:31.443 --> 00:02:34.752
but this is a review of
the permutations formula,

00:02:34.752 --> 00:02:38.475
where people say, "Hey, if I'm saying n,

00:02:38.475 --> 00:02:40.777
"if I'm taking n things,
then I want to figure out

00:02:40.777 --> 00:02:42.298
"how many permutations are there

00:02:42.298 --> 00:02:45.258
"of putting them into let's say k spots,

00:02:45.258 --> 00:02:48.590
"it's going to be equal to n factorial

00:02:48.590 --> 00:02:52.398
"over n minus k factorial."

00:02:52.398 --> 00:02:54.459
That's exactly what we did over here,

00:02:54.459 --> 00:02:57.317
where six is n

00:02:57.317 --> 00:03:00.361
and k, or four is k.

00:03:00.361 --> 00:03:01.512
Four is k.

00:03:01.512 --> 00:03:03.658
Actually let me color code the whole thing

00:03:06.873 --> 00:03:09.218
so that we see the parallel.

00:03:09.219 --> 00:03:10.879
All of that is review.

00:03:10.879 --> 00:03:13.166
Then we went into the
world of combinations.

00:03:13.166 --> 00:03:14.768
In the world of combinations,

00:03:14.768 --> 00:03:17.868
we said permutations make a difference

00:03:17.868 --> 00:03:19.539
between who's sitting in what chair.

00:03:19.539 --> 00:03:21.118
For example, in the permutations world,

00:03:21.118 --> 00:03:22.373
and this is all review, we've covered this

00:03:22.373 --> 00:03:26.133
in the combinations video,
in the permutations world

00:03:26.133 --> 00:03:29.127
A, B, C, D,

00:03:29.127 --> 00:03:33.156
and D, A, B, C,

00:03:33.156 --> 00:03:35.489
these would be two different permutations

00:03:35.489 --> 00:03:37.406
that's being counted in
whatever number this is.

00:03:37.406 --> 00:03:37.959
This is what?

00:03:37.959 --> 00:03:39.831
This is 30 times twelve.

00:03:41.784 --> 00:03:44.720
This is equal to 360.

00:03:46.012 --> 00:03:48.236
Each of these, this is one permutation,

00:03:48.236 --> 00:03:50.521
this is another permutation,

00:03:50.521 --> 00:03:53.495
and if we keep doing it
we would count up to 360.

00:03:53.495 --> 00:03:54.923
But we learned in combinations,

00:03:54.923 --> 00:03:56.031
when we're thinking about combinations,

00:03:56.031 --> 00:03:58.353
let me write combinations.

00:03:58.353 --> 00:04:00.905
If we're saying n choose,

00:04:00.905 --> 00:04:03.195
n choose k,

00:04:03.195 --> 00:04:06.010
or how many combinations are there?

00:04:06.010 --> 00:04:07.770
If we take k things, and
we just wanna figure out

00:04:07.770 --> 00:04:09.203
how many combinatio-

00:04:09.203 --> 00:04:11.384
If we start with n, if
we have pool of n things,

00:04:11.384 --> 00:04:12.557
and we wanna say how many combinations

00:04:12.557 --> 00:04:15.122
of k things are there,
then we would count these

00:04:15.122 --> 00:04:17.363
as the same combination.

00:04:17.363 --> 00:04:20.102
What we really wanna do is we wanna take

00:04:20.102 --> 00:04:22.598
the number of permutations there are.

00:04:22.598 --> 00:04:25.909
We wanna take the number
of permutations there are,

00:04:25.909 --> 00:04:30.909
which is equal to n factorial over

00:04:31.434 --> 00:04:34.042
n minus k factorial,

00:04:34.380 --> 00:04:37.520
over n minus k factorial,

00:04:38.736 --> 00:04:41.324
and we wanna divide by the number of ways

00:04:41.324 --> 00:04:43.065
that you can arrange four people.

00:04:43.065 --> 00:04:44.354
Once again, this takes-

00:04:44.354 --> 00:04:45.151
I remember the first time I learned

00:04:45.151 --> 00:04:46.432
it took my brain a little while.

00:04:46.432 --> 00:04:48.080
If it's taking you a little
while to think about it,

00:04:48.080 --> 00:04:49.079
not a big deal.

00:04:49.079 --> 00:04:51.342
It can be confusing at first,

00:04:51.342 --> 00:04:53.339
but hopefully if you
keep thinking about it

00:04:53.339 --> 00:04:56.880
hopefully you will see
clarity at some moment.

00:04:56.880 --> 00:04:59.492
What we wanna do is we wanna
divide by all of the ways

00:04:59.492 --> 00:05:01.303
that you can arrange four things.

00:05:01.303 --> 00:05:03.265
Cuz once again, in the permutations

00:05:03.265 --> 00:05:05.250
it's counting all of the
different arrangements

00:05:05.250 --> 00:05:07.328
of four things, but we don't wanna count

00:05:07.328 --> 00:05:09.661
all of those different
arrangements of four things.

00:05:09.661 --> 00:05:13.040
We wanna just say they're
all one combination.

00:05:13.040 --> 00:05:15.338
We wanna divide by the number of ways

00:05:15.338 --> 00:05:17.219
to arrange four things.

00:05:17.219 --> 00:05:20.514
Or the number of ways to arrange k things.

00:05:20.514 --> 00:05:21.806
Let me write this down.

00:05:21.806 --> 00:05:23.395
What is the number of ways,

00:05:23.395 --> 00:05:25.670
number of ways,

00:05:26.439 --> 00:05:28.746
to arrange

00:05:29.716 --> 00:05:31.383
k things,

00:05:31.968 --> 00:05:33.415
k things,

00:05:33.415 --> 00:05:34.593
in k spots.

00:05:34.593 --> 00:05:36.374
I encourage you to pause the video,

00:05:36.374 --> 00:05:37.535
because this actually a review

00:05:37.535 --> 00:05:40.053
from the first permutation video.

00:05:40.930 --> 00:05:42.793
If you have k spots, let me do it

00:05:42.793 --> 00:05:45.974
so if this is the first
spot, the second spot,

00:05:45.974 --> 00:05:49.439
third spot, and then
you're gonna go all the way

00:05:49.439 --> 00:05:52.178
to the kth spot.

00:05:52.178 --> 00:05:53.581
For the first spot

00:05:53.581 --> 00:05:55.877
there could be k possibilities.

00:05:55.877 --> 00:05:58.291
There's k things that
could take the first spot.

00:05:58.291 --> 00:05:59.975
For each of those k possibilities,

00:05:59.975 --> 00:06:02.378
how many things can be in the second spot?

00:06:02.378 --> 00:06:04.785
It's gonna be k minus one,
because you already put,

00:06:04.785 --> 00:06:06.839
you already put something
in the first spot.

00:06:06.839 --> 00:06:07.977
Then over here, what's it gonna be?

00:06:07.977 --> 00:06:09.657
K minus two

00:06:09.963 --> 00:06:11.247
all the way to the last spot,

00:06:11.247 --> 00:06:14.002
there's only one thing that
can be put in the last spot.

00:06:14.002 --> 00:06:14.826
What is this thing here?

00:06:14.826 --> 00:06:16.765
K times k minus one times k minus two

00:06:16.765 --> 00:06:20.445
times k minus three all
the way down to one.

00:06:20.445 --> 00:06:23.776
This is just equal to k factorial.

00:06:23.776 --> 00:06:25.843
The number of ways to
arrange k things in k spots,

00:06:25.843 --> 00:06:26.690
k factorial.

00:06:26.690 --> 00:06:29.395
The number of ways to arrange
four things in four spots,

00:06:29.395 --> 00:06:30.811
that's four factorial.

00:06:30.811 --> 00:06:32.634
The number of ways to arrange three things

00:06:32.634 --> 00:06:35.107
in three spots, it's three factorial.

00:06:35.107 --> 00:06:36.879
We could just divide this.

00:06:36.879 --> 00:06:41.515
We could just divide this by k factorial.

00:06:41.515 --> 00:06:44.980
This would get us, this would get us,

00:06:45.273 --> 00:06:47.183
n factorial

00:06:47.901 --> 00:06:51.080
divided by k factorial,

00:06:51.473 --> 00:06:54.088
k factorial

00:06:54.088 --> 00:06:55.768
times,

00:06:55.768 --> 00:07:00.125
times n minus k factorial,

00:07:00.140 --> 00:07:03.492
n minus k,

00:07:03.492 --> 00:07:05.474
n minus k,

00:07:05.474 --> 00:07:07.798
I'll put the factorial right over there.

00:07:07.798 --> 00:07:10.447
This right over here is the formula.

00:07:10.447 --> 00:07:14.234
This right over here is the
formula for combinations.

00:07:14.234 --> 00:07:17.236
Sometimes this is also called
the binomial coefficient.

00:07:17.236 --> 00:07:18.838
People always call this n choose k.

00:07:18.838 --> 00:07:20.394
They'll also write it like this,

00:07:20.394 --> 00:07:22.797
n choose k, especially
when they're thinking

00:07:22.797 --> 00:07:24.840
in terms of binomial coefficients.

00:07:24.840 --> 00:07:27.939
I got into kind of an
abstract tangent here,

00:07:27.939 --> 00:07:30.935
when I started getting
into the general formula.

00:07:30.935 --> 00:07:32.754
Let's go back to our example.

00:07:32.754 --> 00:07:36.112
In our example we saw
that there was 360 ways

00:07:36.112 --> 00:07:40.361
of seating six people into four chairs.

00:07:40.361 --> 00:07:42.625
What if we didn't care about
who's sitting in which chairs

00:07:42.625 --> 00:07:44.947
and just wanna say,
"How many ways are there

00:07:44.947 --> 00:07:47.283
"to choose four people

00:07:47.283 --> 00:07:50.312
"from a pool of six?"

00:07:50.312 --> 00:07:51.652
That would be

00:07:51.652 --> 00:07:53.516
that would be how many ways are there.

00:07:53.516 --> 00:07:55.120
That would be six.

00:07:55.905 --> 00:07:57.942
How many combinations

00:07:57.942 --> 00:08:00.143
if I'm starting with a pool of six,

00:08:00.143 --> 00:08:02.403
how many combinations are there?

00:08:02.403 --> 00:08:04.426
How many combinations are there

00:08:04.426 --> 00:08:06.888
for selecting four?

00:08:06.888 --> 00:08:08.966
Another way of thinking about it is

00:08:08.966 --> 00:08:11.667
how many ways are there to,

00:08:11.667 --> 00:08:14.768
from a pool of six items,

00:08:14.768 --> 00:08:16.330
people in this example,

00:08:16.330 --> 00:08:19.634
how many ways are there
to choose four of them.

00:08:19.634 --> 00:08:23.972
That is going to be, we could do it-

00:08:23.972 --> 00:08:25.439
I'll apply the formula first,

00:08:25.439 --> 00:08:26.299
and then I'll reason through it.

00:08:26.299 --> 00:08:27.591
And like I always say,

00:08:27.591 --> 00:08:28.713
I'm not a huge fan of the formula.

00:08:28.713 --> 00:08:31.380
Every time I revisit it after a few years,

00:08:31.380 --> 00:08:35.295
actually just rethink about it,
as opposed to memorizing it,

00:08:35.295 --> 00:08:37.837
because memorizing is a
good way to not understand

00:08:37.837 --> 00:08:40.264
what's actually going on,

00:08:40.264 --> 00:08:42.086
but if we just apply the formula here,

00:08:42.086 --> 00:08:43.619
I really want you to understand

00:08:43.619 --> 00:08:45.139
what's happening in the formula,

00:08:45.139 --> 00:08:49.222
it would be six factorial
over four factorial,

00:08:49.222 --> 00:08:51.469
over four factorial,

00:08:51.469 --> 00:08:55.164
times six minus four factorial,

00:08:55.164 --> 00:08:58.135
six, oops let me just,

00:08:58.135 --> 00:09:00.888
This is six minus four factorial,

00:09:00.888 --> 00:09:03.156
so this part right over here,

00:09:03.156 --> 00:09:04.348
six minus four fa-

00:09:04.348 --> 00:09:05.403
Let me write it out because I know

00:09:05.403 --> 00:09:07.572
this can be a little bit confusing

00:09:07.572 --> 00:09:09.020
the first time you see it.

00:09:09.020 --> 00:09:12.480
So six minus four factorial,

00:09:13.310 --> 00:09:16.102
factorial, which is equal to,

00:09:16.102 --> 00:09:21.082
which is equal to six
factorial over four factorial,

00:09:21.082 --> 00:09:25.296
over four factorial, times
this thing right over here

00:09:25.296 --> 00:09:28.523
is two factorial, times two factorial,

00:09:28.523 --> 00:09:31.169
which is going to be equal to, we can just

00:09:31.169 --> 00:09:34.907
write out the factorials,
six times five times four

00:09:34.907 --> 00:09:38.484
times three times two times one,

00:09:38.484 --> 00:09:41.142
over four.

00:09:41.804 --> 00:09:45.449
Four time three times two times one

00:09:45.449 --> 00:09:47.172
times,

00:09:47.725 --> 00:09:50.171
times two times one.

00:09:50.171 --> 00:09:53.018
Of course that's going
to cancel with that.

00:09:53.018 --> 00:09:55.340
Then the one doesn't
really change the value,

00:09:55.340 --> 00:09:57.906
so let me get rid of this one here.

00:09:57.906 --> 00:10:02.526
Let's see, this three can
cancel with this three.

00:10:02.526 --> 00:10:05.301
This four could cancel with this four.

00:10:05.301 --> 00:10:09.004
Then it's six divided by
two is going to be three.

00:10:09.004 --> 00:10:13.044
So we are just left with three times five.

00:10:13.044 --> 00:10:14.463
We are left with,

00:10:14.463 --> 00:10:18.176
we are left with, there's
fifteen combinations.

00:10:18.176 --> 00:10:21.275
There's 360 permutations for putting

00:10:21.275 --> 00:10:23.771
six people into four chairs,

00:10:23.771 --> 00:10:25.454
but there's only 15 combinations,

00:10:25.454 --> 00:10:26.870
because we're no longer counting

00:10:26.870 --> 00:10:28.391
all of the different arrangements

00:10:28.391 --> 00:10:30.794
for the same four people
in the four chairs.

00:10:30.794 --> 00:10:32.396
We're saying, "Hey if
it's the same four people,

00:10:32.396 --> 00:10:34.892
"that is now one combination."

00:10:34.892 --> 00:10:36.216
You can see how many ways are there

00:10:36.216 --> 00:10:38.270
to arrange four people into four chairs?

00:10:38.270 --> 00:10:40.755
That's the four factorial
part right over here.

00:10:40.755 --> 00:10:42.508
The four factorial part right over here,

00:10:42.508 --> 00:10:44.435
which is four times three
times two times one,

00:10:44.435 --> 00:10:45.607
which is 24.

00:10:45.607 --> 00:10:47.337
We essentially just took the 360

00:10:47.337 --> 00:10:50.309
divided by 24 to get 15.

00:10:50.309 --> 00:10:54.093
Once again, I don't think
I can stress this enough.

00:10:54.093 --> 00:10:56.206
I wanna make it clear
where this is coming from.

00:10:56.206 --> 00:10:58.532
This right over here,

00:10:58.532 --> 00:10:59.672
let me circle,

00:10:59.672 --> 00:11:02.969
this piece right over here is
the number of permutations.

00:11:02.969 --> 00:11:04.548
This is really just so you can get to

00:11:04.548 --> 00:11:06.742
six times five times four times three,

00:11:06.742 --> 00:11:08.623
which is exactly what we did up here,

00:11:08.623 --> 00:11:09.970
where we reasoned through it.

00:11:09.970 --> 00:11:12.071
Then we just wanna divide
by the number of ways

00:11:12.071 --> 00:11:15.383
you can arrange four items in four spaces.


Handshaking combinations

WEBVTT
Kind: captions
Language: en

00:00:00.328 --> 00:00:02.601
- Let's say that there
are four people in a room.

00:00:02.601 --> 00:00:03.638
And you're probably tired of me

00:00:03.638 --> 00:00:05.143
naming the people with letters,

00:00:05.143 --> 00:00:07.093
but I'm going to continue doing that.

00:00:07.093 --> 00:00:12.093
So the four people in the room
are people A, B, C, and D.

00:00:13.951 --> 00:00:17.297
And they are all told,
"You don't know each other.

00:00:17.297 --> 00:00:19.537
"So I want you to all meet each other.

00:00:19.537 --> 00:00:22.474
"You need to shake the hand, exactly once,

00:00:22.474 --> 00:00:25.875
"of every other person in the
room so that you all meet."

00:00:25.875 --> 00:00:30.616
So my question to you is,
if each of these people

00:00:30.616 --> 00:00:34.989
need to shake the hand of every
other person exactly once,

00:00:34.989 --> 00:00:36.905
how many handshakes are going to occur?

00:00:36.905 --> 00:00:38.886
The number of handshakes
that are going to occur.

00:00:38.886 --> 00:00:41.026
So, like always, pause the video

00:00:41.026 --> 00:00:43.322
and see if you can make sense of this.

00:00:44.704 --> 00:00:47.492
Alright, I'm assuming
you've had a go at it.

00:00:47.492 --> 00:00:49.291
So one way to think about it is,

00:00:49.291 --> 00:00:51.409
if you say there's a handshake,

00:00:52.654 --> 00:00:54.313
two people are party to a handshake.

00:00:54.313 --> 00:00:57.179
We're not talking about
some new three-person

00:00:57.179 --> 00:00:58.792
handshake or four-person handshake,

00:00:58.792 --> 00:01:00.712
we're just talking about the traditional,

00:01:00.712 --> 00:01:02.885
two people shake their right hands.

00:01:02.885 --> 00:01:04.507
And so, there's one person and there's

00:01:04.507 --> 00:01:05.994
another person in this party.

00:01:06.825 --> 00:01:10.026
There's four possibilities of one party.

00:01:10.026 --> 00:01:11.570
And if we assume people aren't
shaking their own hands,

00:01:11.570 --> 00:01:12.868
which we are assuming, they're always

00:01:12.868 --> 00:01:14.569
going to shake someone else's hand.

00:01:16.968 --> 00:01:19.417
For each of these four
possibilities who's this party,

00:01:19.417 --> 00:01:22.737
there's three possibilities
who's the other party.

00:01:22.737 --> 00:01:24.503
And so you might say that there's

00:01:24.503 --> 00:01:27.668
four times three handshakes.

00:01:28.237 --> 00:01:32.662
Since there's four times
three possible handshakes.

00:01:32.662 --> 00:01:34.087
And what I'd like you to
do is think a little bit

00:01:34.087 --> 00:01:36.599
about whether this is
right, whether there would

00:01:36.599 --> 00:01:40.073
actually be 12 handshakes.

00:01:41.489 --> 00:01:43.143
You might have thought
about it, and you might say,

00:01:43.143 --> 00:01:46.934
there's four times three, this is actually

00:01:46.934 --> 00:01:48.418
counting the permutations.

00:01:48.418 --> 00:01:50.564
This is counting how
many ways can you permute

00:01:50.564 --> 00:01:54.933
four people into two
buckets, the two buckets

00:01:54.933 --> 00:01:57.401
of handshakers, where you care about

00:01:57.401 --> 00:01:58.564
which bucket they are in.

00:01:58.564 --> 00:02:00.896
Whether they're handshaker number one,

00:02:00.896 --> 00:02:02.881
or handshaker number two.

00:02:02.881 --> 00:02:07.881
This would count A being
the number one handshaker

00:02:07.994 --> 00:02:10.109
and B being the number two handshaker

00:02:10.109 --> 00:02:13.830
as being different than B
being the number one handshaker

00:02:13.830 --> 00:02:16.797
and A being the number two handshaker.

00:02:16.797 --> 00:02:18.526
But we don't want both
of these things to occur.

00:02:18.526 --> 00:02:20.840
We don't want A to shake B's hand,

00:02:20.840 --> 00:02:23.952
where A is facing north
and B is facing south.

00:02:23.952 --> 00:02:26.510
And then another time,
they shake hands again

00:02:26.510 --> 00:02:29.188
where now B is facing north
and A is facing south.

00:02:29.188 --> 00:02:30.591
We only have to do it once.

00:02:30.591 --> 00:02:33.261
These are actually the same thing,

00:02:33.261 --> 00:02:36.013
so there's no reason for
both of these to occur.

00:02:36.013 --> 00:02:38.602
So we are going to be double counting.

00:02:38.602 --> 00:02:42.337
So what we really want to do
is think about combinations.

00:02:42.337 --> 00:02:45.196
One way to think about it
is, you have four people.

00:02:45.196 --> 00:02:48.760
In a world of four people
or a pool of four people,

00:02:48.760 --> 00:02:51.081
how many ways can you choose two?

00:03:01.387 --> 00:03:02.247
Because that's what we're doing.

00:03:02.247 --> 00:03:04.325
Each handshake is just really a selection

00:03:04.325 --> 00:03:05.835
of two of these people.

00:03:05.835 --> 00:03:07.204
And so we want to say, how many ways

00:03:07.204 --> 00:03:08.935
can we select two people?

00:03:08.935 --> 00:03:12.625
So that each combination,
each of these ways

00:03:12.625 --> 00:03:14.500
to select two people
should have a different

00:03:14.500 --> 00:03:16.166
combination of people in it.

00:03:16.166 --> 00:03:19.362
If two of them have the same, AB and BA,

00:03:19.362 --> 00:03:21.283
these are the same combination.

00:03:21.283 --> 00:03:24.211
And so this is really
a combinations problem.

00:03:24.211 --> 00:03:26.509
This is really equivalent to saying,

00:03:26.509 --> 00:03:29.842
how many ways are there
to choose two people

00:03:29.842 --> 00:03:31.398
from a pool of four?

00:03:31.398 --> 00:03:33.173
Or four choose two.

00:03:35.925 --> 00:03:39.669
And so this is going to
be, well how many ways

00:03:39.669 --> 00:03:44.236
are there to permute four
people into three spots?

00:03:44.236 --> 00:03:47.187
Which is going to be four times three.

00:03:47.187 --> 00:03:49.843
Which we just figured out
right over there, which is 12.

00:03:49.843 --> 00:03:51.110
Actually want to do it in that green color

00:03:51.110 --> 00:03:53.104
so you see where that came from.

00:03:53.942 --> 00:03:56.288
So four times three.

00:03:56.288 --> 00:03:57.424
And then you're going to divide that

00:03:57.424 --> 00:04:02.045
by the number of ways you
can arrange two people.

00:04:02.045 --> 00:04:03.504
Well you can arrange two
people in two different ways.

00:04:03.504 --> 00:04:04.403
One's on the left, one's on the right,

00:04:04.403 --> 00:04:05.389
or the other one's on the left

00:04:05.389 --> 00:04:06.700
or the other one's on the right.

00:04:06.700 --> 00:04:09.010
Or, you could also view
that as two factorial,

00:04:09.010 --> 00:04:10.600
which is also equal to two.

00:04:10.600 --> 00:04:14.282
So we could write this down as two.

00:04:14.282 --> 00:04:19.282
This is the number of ways
to arrange two people.

00:04:33.190 --> 00:04:35.695
This up here, that's the permutations.

00:04:36.664 --> 00:04:39.020
That's the number of permutations

00:04:39.020 --> 00:04:41.547
if you take two people
from a pool of four.

00:04:41.547 --> 00:04:43.709
So here you would care about order.

00:04:44.248 --> 00:04:46.440
And so one way to think about it, this two

00:04:46.440 --> 00:04:49.793
is correcting for this
double counting here.

00:04:49.793 --> 00:04:51.733
And if you want to apply
the formula, you could.

00:04:51.733 --> 00:04:54.134
I just kind of reasoned through it again.

00:04:54.134 --> 00:04:55.596
You could literally say,
four times three is 12.

00:04:55.596 --> 00:04:57.292
We're double counting because there's

00:04:57.292 --> 00:04:58.752
two ways to arrange two people.

00:04:58.752 --> 00:05:00.972
So you just divide it by two.

00:05:03.418 --> 00:05:05.729
And then you are going
to be left with six.

00:05:05.729 --> 00:05:08.090
You can think of it in terms of this,

00:05:08.090 --> 00:05:09.932
or you could just apply the formula.

00:05:09.932 --> 00:05:13.569
You could just say, four choose two,

00:05:13.569 --> 00:05:15.705
or the number of combinations of selecting

00:05:15.705 --> 00:05:17.447
two from a group of four.

00:05:17.447 --> 00:05:21.880
This is going to be four
factorial over two factorial

00:05:21.880 --> 00:05:25.338
times four minus two factorial.

00:05:25.338 --> 00:05:26.918
And I'm going to make this
color different just so

00:05:26.918 --> 00:05:30.561
you can keep track of how
I'm at least applying this.

00:05:30.561 --> 00:05:32.145
And so what is this going to be?

00:05:32.145 --> 00:05:36.984
This is going to be four times
three times two times one,

00:05:36.984 --> 00:05:41.984
over two times one times
this right over here is

00:05:45.340 --> 00:05:46.946
two times one.

00:05:46.946 --> 00:05:49.196
So that would cancel with that.

00:05:49.196 --> 00:05:52.307
Four divided by two is two.

00:05:52.307 --> 00:05:56.208
Two times three divided
by one is equal to six.

00:05:56.208 --> 00:05:58.167
And to just really hit the point home,

00:05:58.167 --> 00:06:00.349
let's actually draw it out.

00:06:02.591 --> 00:06:05.356
A could shake B's hand.

00:06:05.356 --> 00:06:07.248
A could shake C's hand.

00:06:07.248 --> 00:06:09.160
A could shake D's hand.

00:06:11.540 --> 00:06:13.781
Let me just do what we
calculated first, the 12.

00:06:13.781 --> 00:06:15.665
B could shake A's hand.

00:06:15.665 --> 00:06:17.267
B could shake C's hand.

00:06:17.267 --> 00:06:21.853
B could shake D's hand.

00:06:22.299 --> 00:06:24.395
C could shake A's hand.

00:06:24.395 --> 00:06:26.252
C could shake B's hand.

00:06:26.252 --> 00:06:30.304
C could shake D's hand.

00:06:30.304 --> 00:06:31.906
D could shake A's hand.

00:06:31.906 --> 00:06:33.729
D could shake B's hand.

00:06:33.729 --> 00:06:36.086
D could shake C's hand.

00:06:36.086 --> 00:06:37.712
And this is 12 right over here,

00:06:37.712 --> 00:06:38.906
and this is the permutations.

00:06:38.906 --> 00:06:41.507
If D shaking C's hand
was actually different

00:06:41.507 --> 00:06:45.145
than C shaking D's hand,
then we would count 12.

00:06:45.145 --> 00:06:47.228
But we just wanted to say, how many ways,

00:06:47.228 --> 00:06:49.146
they just have to meet each other once.

00:06:49.146 --> 00:06:50.748
And so we're double counting.

00:06:50.748 --> 00:06:54.669
So AB is the same thing as BA.

00:06:54.669 --> 00:06:56.510
AC is the same thing as CA.

00:06:56.510 --> 00:06:58.964
AD is the same thing as DA.

00:06:59.579 --> 00:07:02.172
BC is the same thing as CB.

00:07:02.172 --> 00:07:04.803
BD is the same thing as DB.

00:07:04.803 --> 00:07:07.325
CD is the same thing as DC.

00:07:07.325 --> 00:07:08.823
And so we would be left with,

00:07:08.823 --> 00:07:11.551
if we correct for the double counting,

00:07:11.551 --> 00:07:16.551
we're left with one,
two, three, four, five,

00:07:16.624 --> 00:07:18.401
six combinations.

00:07:18.401 --> 00:07:23.401
Six possible ways of choosing
two from a pool of four.

00:07:23.949 --> 00:07:25.011
Especially when you don't care about

00:07:25.011 --> 00:07:27.268
the order in which you choose them.


Possible three letter words

WEBVTT
Kind: captions
Language: en

00:00:00.303 --> 00:00:02.182
- [Voiceover] So let's ask ourselves some

00:00:02.182 --> 00:00:05.480
interesting questions about
alphabets in the English language.

00:00:05.480 --> 00:00:07.199
And in case you don't remember

00:00:07.199 --> 00:00:09.846
and are in the mood to count,
there are 26 alphabets.

00:00:09.846 --> 00:00:13.003
So if you go, "A, B, C,
D, E, F, G, H, I, J, K,

00:00:13.003 --> 00:00:15.836
L, M, N, O, P, Q, R, S,
T, U, V, W, X, Y, and Z,"

00:00:15.836 --> 00:00:20.288
you'll get, you'll get 26, 26 alphabets.

00:00:20.288 --> 00:00:22.887
Now let's ask some interesting questions.

00:00:22.887 --> 00:00:24.884
So given that there are 26 alphabets

00:00:24.884 --> 00:00:26.912
in the English language, how many

00:00:26.912 --> 00:00:31.315
possible three letter words are there?

00:00:31.315 --> 00:00:33.637
And we're not going to be
thinking about phonetics

00:00:33.637 --> 00:00:35.317
or how hard it is to pronounce it.

00:00:35.317 --> 00:00:39.226
So, for example, the word, the word ZGT

00:00:39.226 --> 00:00:42.042
would be a legitimate
word in this example.

00:00:42.042 --> 00:00:47.042
Or the word, the word, the word SKJ

00:00:47.507 --> 00:00:50.147
would be a legitimate
word in this example.

00:00:50.147 --> 00:00:52.824
So how many possible three letter words

00:00:52.824 --> 00:00:54.728
are there in the English language?

00:00:54.728 --> 00:00:55.774
I encourage you to pause the video

00:00:55.774 --> 00:00:57.376
and try to think about it.

00:00:57.377 --> 00:01:00.107
Alright, I assume you've had a go at it.

00:01:00.107 --> 00:01:02.122
So let's just think about
it, for three letter words

00:01:02.122 --> 00:01:04.944
there's three spaces, so how many

00:01:04.944 --> 00:01:07.522
possibilities are there for the first one?

00:01:07.522 --> 00:01:10.842
Well, there's 26 possible
letters for the first one.

00:01:10.842 --> 00:01:13.466
Anything from a to z
would be completely fine.

00:01:13.466 --> 00:01:16.020
Now how many possibilities
for the second one?

00:01:16.020 --> 00:01:17.738
And I intentionally ask this to you

00:01:17.738 --> 00:01:21.639
to be a distractor because
we've seen a lot of examples.

00:01:21.639 --> 00:01:23.287
We're saying, "Oh,
there's 26 possibilities

00:01:23.287 --> 00:01:25.958
"for the first one and maybe
there's 25 for the second one,

00:01:25.958 --> 00:01:28.396
"and then 24 for the third,"
but that's not the case

00:01:28.396 --> 00:01:31.477
right over here because
we can repeat letters.

00:01:31.477 --> 00:01:33.922
I didn't say that all of the
letters had to be different.

00:01:33.922 --> 00:01:38.922
So, for example, the word, the word HHH

00:01:39.517 --> 00:01:43.627
would also be a legitimate word

00:01:43.627 --> 00:01:45.415
in our example right over here.

00:01:45.415 --> 00:01:47.923
So we have 26 possibilities
for the second letter

00:01:47.923 --> 00:01:50.639
and we have 26 possibilities
for the third letter.

00:01:50.639 --> 00:01:52.915
So we're going to have, and
I don't know what this is,

00:01:52.915 --> 00:01:55.585
26 to the third power possibilities,

00:01:55.585 --> 00:01:58.952
or 26 times 26 time 26 and you
can figure out what that is.

00:01:58.952 --> 00:02:00.895
That is how many possible three letter

00:02:00.895 --> 00:02:03.117
words we can have for the English language

00:02:03.117 --> 00:02:05.082
if we didn't care about
how to pronounceable

00:02:05.082 --> 00:02:08.541
they are, if they meant anything
and if we repeated letters.

00:02:08.541 --> 00:02:10.375
Now let's ask a different question.

00:02:10.375 --> 00:02:13.440
What if we said, "How many
possible three letter words

00:02:13.440 --> 00:02:17.674
"are there if we want
all different letters?"

00:02:17.674 --> 00:02:20.940
So we want all different letters.

00:02:20.940 --> 00:02:24.168
So these all have to be different letters.

00:02:24.168 --> 00:02:27.302
Different, different
letters and once again,

00:02:27.302 --> 00:02:30.739
pause the video and see if
you can think it through.

00:02:30.739 --> 00:02:34.175
Alright, so this is where
permutations start to be useful.

00:02:34.175 --> 00:02:36.659
Although, I think a lot
of things like this,

00:02:36.659 --> 00:02:38.145
it's always best to reason through

00:02:38.145 --> 00:02:41.753
than try to figure out if
some formula applies to it.

00:02:41.753 --> 00:02:44.043
So in this situation,
well, if we went in order,

00:02:44.043 --> 00:02:46.806
we could have 26 different
letters for the first one,

00:02:46.806 --> 00:02:49.075
26 different possibilities
for the first one.

00:02:49.075 --> 00:02:50.305
You know, I'm always
starting with that one,

00:02:50.305 --> 00:02:52.248
but there's nothing special
about the one on the left.

00:02:52.248 --> 00:02:54.213
We could say that the one on the right,

00:02:54.213 --> 00:02:56.899
there's 26 possibilities, well for each

00:02:56.899 --> 00:03:00.087
of those possibilities, for
each of those 26 possibilities,

00:03:00.087 --> 00:03:02.804
there might be 25 possibilities

00:03:02.804 --> 00:03:04.197
for what we put in the middle one

00:03:04.197 --> 00:03:06.333
if we say we're going to
figure out the middle one next.

00:03:06.333 --> 00:03:09.360
And then for each of these
25 times 26 possibilities

00:03:09.360 --> 00:03:11.465
for where we figured
out two of the letters,

00:03:11.465 --> 00:03:13.647
there's 24 possibilities
because we've already used

00:03:13.647 --> 00:03:16.914
two letters for the last
bucket that we haven't filled.

00:03:16.914 --> 00:03:19.740
And the only reason I went 26, 25, 24

00:03:19.740 --> 00:03:21.806
is to show you there's nothing
special about always filling

00:03:21.806 --> 00:03:26.155
in the left most letter or
the left most chair first.

00:03:26.155 --> 00:03:28.014
It's just about, well,
let's just think in terms

00:03:28.014 --> 00:03:30.783
of let's just fill out
one of the buckets first.

00:03:30.783 --> 00:03:32.362
Hey, we have the most
possibilities for that.

00:03:32.362 --> 00:03:34.800
Once we use something up,
then for each of those

00:03:34.800 --> 00:03:37.307
possibilities we'll
have one left, one less

00:03:37.307 --> 00:03:40.790
for the next, the next bucket.

00:03:40.790 --> 00:03:43.577
And so I could do 24 times 25 times 26,

00:03:43.577 --> 00:03:44.784
but just so I don't fully confuse you,

00:03:44.784 --> 00:03:46.688
I'll go back to what I have been doing.

00:03:46.688 --> 00:03:49.242
26 possibilities for the left most one.

00:03:49.242 --> 00:03:50.918
For each of those, you
would have 25 possibilities

00:03:50.918 --> 00:03:53.723
for the next one that you're
going to try to figure out

00:03:53.723 --> 00:03:54.931
because you already used one letter

00:03:54.931 --> 00:03:56.092
and they have to be different.

00:03:56.092 --> 00:03:58.413
And then for the last
bucket, you're going to have

00:03:58.413 --> 00:04:01.269
24 possibilities, so this is going to be

00:04:01.269 --> 00:04:05.853
26 times 25 times 24,

00:04:05.853 --> 00:04:07.222
whatever that happens to be.

00:04:07.222 --> 00:04:09.350
And if we wanted to
write it in the notation

00:04:09.350 --> 00:04:12.670
of permutations, we would
say that this is equal to,

00:04:12.670 --> 00:04:16.199
we're taking 26 things, sorry, not two p.

00:04:16.199 --> 00:04:18.328
20, my brain is malfunctioning.

00:04:18.328 --> 00:04:21.098
26, we're figuring out how
many permutations are there

00:04:21.098 --> 00:04:24.256
for putting 26 different things into

00:04:24.256 --> 00:04:27.414
three different spaces

00:04:27.414 --> 00:04:30.572
and this is 26, if we just blindly apply

00:04:30.572 --> 00:04:32.963
the formula, which I never suggest doing.

00:04:32.963 --> 00:04:36.642
It would be 26 factorial over 26

00:04:36.642 --> 00:04:39.643
minus three factorial,

00:04:39.643 --> 00:04:43.115
which would be 26 factorial

00:04:43.115 --> 00:04:45.267
over 23 factorial,

00:04:45.267 --> 00:04:47.771
which is going to be
exactly this right over here

00:04:47.771 --> 00:04:51.005
because the 23 times 22 times
21 all the way down to one

00:04:51.005 --> 00:04:54.116
is going to cancel with the 23 factorial.

00:04:54.116 --> 00:04:55.973
And so the whole point of this video,

00:04:55.973 --> 00:04:58.945
there's two points, is one,
as soon as someone says,

00:04:58.945 --> 00:05:00.617
"How many different
letters could you form"

00:05:00.617 --> 00:05:01.639
or something like that, you don't just

00:05:01.639 --> 00:05:03.891
blindly do permutations or combinations.

00:05:03.891 --> 00:05:06.213
You think about well, what is
being asked in the question.

00:05:06.213 --> 00:05:09.998
Here, I really just have to
take 26 times 26 times 26.

00:05:09.998 --> 00:05:11.168
The other thing I want to point out,

00:05:11.168 --> 00:05:12.529
and I know I keep pointing it out,

00:05:12.529 --> 00:05:14.781
and it's probably getting tiring to you,

00:05:14.781 --> 00:05:18.682
is even when permutations are applicable,

00:05:18.682 --> 00:05:20.911
in my brain, at least,
it's always more valuable

00:05:20.911 --> 00:05:23.186
to just try to reason through the problem

00:05:23.186 --> 00:05:25.972
as opposed to just saying,
"Oh there's this formula

00:05:25.972 --> 00:05:28.155
"that I remember from weeks or years ago

00:05:28.155 --> 00:05:31.848
"in my life that had an N
factorial and K factorial

00:05:31.848 --> 00:05:33.666
"and I had to memorize
it, I have to look it up."

00:05:33.666 --> 00:05:37.696
Always much more useful
to just reason it through.


Zero factorial or 0!

WEBVTT
Kind: captions
Language: en

00:00:00.182 --> 00:00:03.074
- If you've been paying
particularly close attention to

00:00:03.443 --> 00:00:06.404
our use of the factorial
operator in the videos

00:00:06.404 --> 00:00:09.121
on permutations and
combinations, you may or may not

00:00:09.121 --> 00:00:11.745
have noticed something
that might be interesting.

00:00:11.745 --> 00:00:13.621
So let's just review
factorial a little bit.

00:00:13.621 --> 00:00:17.727
So if I were to say n
factorial, that of course

00:00:17.727 --> 00:00:22.727
is going to be n times n
minus one times n minus two,

00:00:28.759 --> 00:00:33.625
and I would just keep going
down until I go to times one.

00:00:33.625 --> 00:00:36.292
So I would keep decrementing
n until I get to one,

00:00:36.292 --> 00:00:39.260
and then I would multiply
all those things together.

00:00:39.260 --> 00:00:41.010
So, for example, and
all of this is review.

00:00:41.010 --> 00:00:43.131
If I were to say three factorial,

00:00:43.131 --> 00:00:46.786
that's going to be three
times two times one.

00:00:46.786 --> 00:00:48.919
If I were to say two factorial,

00:00:48.919 --> 00:00:51.924
that's going to be two times one.

00:00:51.924 --> 00:00:55.340
One factorial, by that logic, I just keep

00:00:55.340 --> 00:00:56.775
decrementing until I get
to one, but I don't even

00:00:56.775 --> 00:00:58.961
have to decrement here,
I'm already at one.

00:00:58.961 --> 00:01:01.093
So I just multiply one.

00:01:01.093 --> 00:01:04.684
Now what about zero factorial,
this is interesting.

00:01:04.684 --> 00:01:07.642
Zero factorial.

00:01:08.167 --> 00:01:11.428
So one logical thing is to say,

00:01:11.428 --> 00:01:14.656
maybe zero factorial is zero.

00:01:15.209 --> 00:01:17.975
I'm just starting with itself,
it's already below one.

00:01:18.388 --> 00:01:21.661
Maybe it is zero.

00:01:21.661 --> 00:01:24.999
Now what we will see is
that this is not the case

00:01:24.999 --> 00:01:28.792
that mathematics,
mathematicians have decided.

00:01:28.792 --> 00:01:30.970
This is what's interesting,
the factorial operation,

00:01:30.970 --> 00:01:34.314
this is something that
humans have invented,

00:01:34.314 --> 00:01:36.053
that they think is just
an interesting thing,

00:01:36.053 --> 00:01:37.596
it's a useful notation.

00:01:37.596 --> 00:01:39.999
So they can define what it does.

00:01:39.999 --> 00:01:43.713
And mathematicians have
found it far more useful

00:01:43.713 --> 00:01:47.604
to define zero factorial
as something else.

00:01:47.604 --> 00:01:50.574
To define zero factorial as...

00:01:50.574 --> 00:01:52.618
And there's a little
bit of a drumroll here.

00:01:52.618 --> 00:01:55.880
They believe zero factorial should be one.

00:01:55.880 --> 00:01:58.537
And I know, based on the reasoning,

00:01:58.537 --> 00:02:00.361
the conceptual reasoning of this,

00:02:00.361 --> 00:02:02.055
this doesn't make any sense.

00:02:02.055 --> 00:02:04.099
But since we've already
been exposed a little bit

00:02:04.099 --> 00:02:09.099
to permutations, I'll show you
why this is a useful concept,

00:02:09.200 --> 00:02:12.447
especially in the world of
permutations and combinations.

00:02:12.447 --> 00:02:16.276
Which is, frankly, where
factorial shows up the most.

00:02:17.552 --> 00:02:18.878
Most of the cases that
I've ever seen factorial

00:02:18.878 --> 00:02:20.704
in anything has been in the situations

00:02:20.704 --> 00:02:22.278
of permutations and combinations.

00:02:22.278 --> 00:02:23.334
And in a few other things,

00:02:23.334 --> 00:02:25.654
but mainly permutations and combinations.

00:02:25.654 --> 00:02:27.802
So let's review a little bit.

00:02:27.802 --> 00:02:31.359
We've said that if we
have n things and we want

00:02:31.359 --> 00:02:33.134
to figure out the number
of ways to permute them

00:02:33.134 --> 00:02:38.134
into k spaces, it's
going to be n factorial

00:02:39.680 --> 00:02:44.680
over n minus k factorial.

00:02:45.973 --> 00:02:50.466
Now we've also said
that if we had n things

00:02:50.466 --> 00:02:53.438
that we want to permute into n places,

00:02:53.438 --> 00:02:56.217
well this really should
just be n factorial.

00:02:57.257 --> 00:02:58.516
Let's just do this.

00:02:58.542 --> 00:03:02.609
This is the first place,
this is the second place,

00:03:02.609 --> 00:03:04.358
this is the third place.

00:03:04.358 --> 00:03:08.421
All the way, you get to the nth place.

00:03:09.113 --> 00:03:11.950
There would be n possibilities for

00:03:11.950 --> 00:03:13.327
who's in the first position,

00:03:13.327 --> 00:03:15.610
or which object is in the first position.

00:03:15.610 --> 00:03:17.014
And then for each of those possibilities,

00:03:17.014 --> 00:03:18.873
there would be n minus one possibilities

00:03:18.873 --> 00:03:21.751
for which object you choose
to put in the second position,

00:03:21.751 --> 00:03:23.932
because you've already put
one into that position.

00:03:23.932 --> 00:03:26.993
Now for each of these n times
n minus one possibilities,

00:03:26.993 --> 00:03:29.066
where you've placed two
things, there would be

00:03:29.066 --> 00:03:31.500
n minus two possibilities of what goes

00:03:31.500 --> 00:03:33.396
in the third position,
and then you would just

00:03:33.396 --> 00:03:35.316
go all the way down to one.

00:03:35.316 --> 00:03:36.148
And this thing right over here

00:03:36.148 --> 00:03:38.167
is exactly what we wrote over here.

00:03:38.167 --> 00:03:43.167
This is equal to n factorial.

00:03:43.867 --> 00:03:46.549
But if we directly applied this formula,

00:03:47.963 --> 00:03:50.564
this would need to be n factorial

00:03:50.934 --> 00:03:54.859
over n minus n factorial.

00:03:54.859 --> 00:03:56.824
Then you might see why
this is interesting.

00:03:56.824 --> 00:04:01.824
Because this is going to be n
factorial over zero factorial.

00:04:03.429 --> 00:04:07.636
So in order for this formula to apply,

00:04:07.636 --> 00:04:10.440
even in the case where k is equal to n,

00:04:13.516 --> 00:04:15.363
which is this one right over here,

00:04:15.363 --> 00:04:18.926
and for that to be consistent
with just plain old logic,

00:04:18.926 --> 00:04:22.398
zero factorial needs to be equal to one.

00:04:22.398 --> 00:04:24.824
And so the mathematics
community has decided,

00:04:24.824 --> 00:04:27.467
this thing we've constructed
called factorial,

00:04:27.467 --> 00:04:30.058
we said you put an exclamation
mark behind something,

00:04:30.058 --> 00:04:31.569
in all of our heads, we say you kind of

00:04:31.569 --> 00:04:33.427
count down that number all the way to one

00:04:33.427 --> 00:04:36.105
and you keep multiplying them for zero,

00:04:36.105 --> 00:04:37.456
we're just going to define this.

00:04:37.456 --> 00:04:38.709
We're just going to define,

00:04:38.709 --> 00:04:40.450
make a mathematical definition.

00:04:40.450 --> 00:04:45.450
We're just going to say zero
factorial is equal to one.

00:04:46.139 --> 00:04:47.727
And it's actually quite useful.


Factorial and counting seat arrangements

WEBVTT
Kind: captions
Language: en

00:00:00.243 --> 00:00:02.683
- So, let's say that we have a round table

00:00:02.683 --> 00:00:05.083
and we have three chairs
around that round table.

00:00:05.083 --> 00:00:06.903
This is one chair right over here,

00:00:06.903 --> 00:00:08.443
this is another chair,

00:00:08.443 --> 00:00:10.543
and that is another chair.

00:00:10.543 --> 00:00:12.103
We can number the chairs;

00:00:12.103 --> 00:00:13.203
that is chair one,

00:00:13.203 --> 00:00:14.483
that is chair two,

00:00:14.483 --> 00:00:16.023
and that is chair three.

00:00:16.023 --> 00:00:17.943
Now let's assume that
there are three people

00:00:17.943 --> 00:00:20.993
who want to sit in these three chairs,

00:00:20.993 --> 00:00:22.869
so there is Person A,

00:00:22.869 --> 00:00:24.469
there is Person B,

00:00:24.469 --> 00:00:26.229
and there is Person C.

00:00:26.229 --> 00:00:29.049
What we want to do is we want to count

00:00:29.049 --> 00:00:31.549
the number of ways that these three people

00:00:31.549 --> 00:00:33.249
can sit in these three chairs.

00:00:33.249 --> 00:00:34.669
What do I mean by that?

00:00:34.669 --> 00:00:39.289
Well, if A sits in chair
one, B sits in chair three,

00:00:39.289 --> 00:00:42.822
and C sits in chair two, that is one

00:00:42.822 --> 00:00:44.762
scenario right over there.

00:00:44.762 --> 00:00:47.702
So this is one of the possible scenarios,

00:00:47.702 --> 00:00:50.622
A in seat one, C in seat
two, B in seat three.

00:00:50.622 --> 00:00:54.421
Another scenario could be B in seat one,

00:00:56.235 --> 00:01:00.402
maybe C still sits in seat
two, C still sits there,

00:01:00.402 --> 00:01:03.282
and now A would be in seat three.

00:01:03.282 --> 00:01:05.361
So this is one scenario and two scenarios.

00:01:05.361 --> 00:01:07.142
My question to you is how many

00:01:07.142 --> 00:01:09.122
scenarios are there like this?

00:01:09.122 --> 00:01:10.362
I encourage you to pause the video

00:01:10.362 --> 00:01:13.399
and try to think through it.

00:01:13.399 --> 00:01:14.847
I'm assuming you've had a go at it,

00:01:14.847 --> 00:01:15.947
now let's work through it together.

00:01:15.947 --> 00:01:17.467
I want to do it very systematically

00:01:17.467 --> 00:01:21.042
so that we don't forget
any of the scenarios.

00:01:21.042 --> 00:01:22.314
The way I'm going to
do it is, I'm going to

00:01:22.314 --> 00:01:24.494
create three blanks that
represent each of the chairs.

00:01:24.494 --> 00:01:26.134
So here, it's in a circle,
but we could just say

00:01:26.134 --> 00:01:29.174
that blank is chair one,
that's the blank for chair two,

00:01:29.174 --> 00:01:31.794
and that is the blank for chair three.

00:01:31.794 --> 00:01:34.089
I'm going to start with seat
one, and I'm going to say,

00:01:34.089 --> 00:01:35.814
"Well, what are the different scenarios?"

00:01:35.814 --> 00:01:38.194
A could sit there, B could
sit there, C could sit there.

00:01:38.194 --> 00:01:40.614
Then for each of those, figure
out who could sit in seat two

00:01:40.614 --> 00:01:41.754
and then for each of those,

00:01:41.754 --> 00:01:43.974
figure out who could sit in seat three.

00:01:43.974 --> 00:01:45.174
So let's do that.

00:01:45.174 --> 00:01:49.961
First, what are the scenarios
of who could sit in seat one?

00:01:51.099 --> 00:01:53.316
Maybe I'll write it this way,

00:01:53.316 --> 00:01:56.189
maybe I'll do seat one.

00:01:56.189 --> 00:02:00.658
So right now, we're only
going to fill seat one.

00:02:00.658 --> 00:02:03.016
Well A could sit in seat one,

00:02:03.016 --> 00:02:07.644
in which case we haven't filled
seat two or seat three yet.

00:02:07.644 --> 00:02:09.442
So that's two or three.

00:02:09.442 --> 00:02:12.322
B could sit in seat one,
and we haven't figured out

00:02:12.322 --> 00:02:16.077
who sits in seat two or seat three yet.

00:02:16.077 --> 00:02:18.882
Then we could have C sitting in seat one,

00:02:18.882 --> 00:02:21.214
and we still have to figure
out who's going to sit

00:02:21.214 --> 00:02:24.278
in seat two and seat three.

00:02:24.278 --> 00:02:26.071
For each of these, let's figure out who

00:02:26.071 --> 00:02:27.971
could be sitting in seat two.

00:02:27.971 --> 00:02:32.611
So, seats one and two.

00:02:33.903 --> 00:02:36.132
For this one right over here,

00:02:37.824 --> 00:02:41.194
we have A in seat one for sure.

00:02:41.194 --> 00:02:43.520
We could either have
B sitting in seat two,

00:02:43.520 --> 00:02:47.080
in which case, I'll leave this
blank although you can use

00:02:47.080 --> 00:02:48.660
a little bit of deductive
reasoning to figure out

00:02:48.660 --> 00:02:50.500
who's going to be sitting in seat three.

00:02:50.500 --> 00:02:52.080
So I'll leave seat three blank.

00:02:52.080 --> 00:02:56.640
Or, you could have C sitting in seat two.

00:02:56.640 --> 00:02:59.620
We still haven't figured out
who's sitting in seat three,

00:02:59.620 --> 00:03:00.920
although once again, a little bit

00:03:00.920 --> 00:03:02.900
of deductive reasoning might tell us.

00:03:02.900 --> 00:03:06.994
So these are the scenarios
where A is in seat one.

00:03:06.994 --> 00:03:09.370
Now what about when B is in seat one?

00:03:09.370 --> 00:03:12.804
We could put A in seat two, and we still

00:03:12.804 --> 00:03:14.984
need to figure out who's in seat three.

00:03:14.984 --> 00:03:17.984
Or, we could put C in seat two,

00:03:17.984 --> 00:03:19.104
and we still need to figure out

00:03:19.104 --> 00:03:22.224
who's going to be in seat three.

00:03:22.224 --> 00:03:26.464
Finally, let's look at C,
where C is in seat one.

00:03:26.464 --> 00:03:31.464
You could either put A in seat two or

00:03:31.870 --> 00:03:35.757
you could put B in seat two.

00:03:36.510 --> 00:03:38.384
Now let's fill out all three of the seats.

00:03:38.384 --> 00:03:42.863
So seats one, two,

00:03:42.863 --> 00:03:44.304
and three.

00:03:44.304 --> 00:03:47.024
For this scenario right over here,

00:03:47.024 --> 00:03:49.204
there's only one option that
you could put in seat three,

00:03:49.204 --> 00:03:52.177
the only person who hasn't
sat down yet is person C.

00:03:52.177 --> 00:03:55.070
So this will be A, B, C.

00:03:55.070 --> 00:03:55.854
And what would this be?

00:03:55.854 --> 00:03:57.180
The only person who hasn't sat down

00:03:57.180 --> 00:03:58.916
in this scenario yet to fill this chair,

00:03:58.916 --> 00:04:00.336
the only option for this chair,

00:04:00.336 --> 00:04:01.776
there's only one option here,

00:04:01.776 --> 00:04:03.576
it's going to be person B.

00:04:03.576 --> 00:04:06.236
It's A, C, B.

00:04:06.236 --> 00:04:09.436
This one is going to be B, A, C.

00:04:09.436 --> 00:04:11.016
C's the only person who could sit there,

00:04:11.016 --> 00:04:12.836
A is the only person who could sit there.

00:04:12.836 --> 00:04:14.776
B, C, A.

00:04:14.776 --> 00:04:17.656
And then here, B is the only person

00:04:17.656 --> 00:04:19.216
who could sit in seat three.

00:04:19.216 --> 00:04:20.676
And here A is the only person

00:04:20.676 --> 00:04:22.616
who could sit in seat three.

00:04:22.616 --> 00:04:24.136
How many scenarios do we have?

00:04:24.136 --> 00:04:28.346
We have one, two, three, four,

00:04:28.346 --> 00:04:31.234
five, and six.

00:04:31.234 --> 00:04:32.476
And six is our answer.

00:04:32.476 --> 00:04:34.396
That's the number of
scenarios that you could have

00:04:34.396 --> 00:04:37.469
the different people sitting
in the different chairs.

00:04:37.469 --> 00:04:38.767
Now, I know what you're thinking,

00:04:38.767 --> 00:04:40.456
"Okay Sal, this was a
scenario with three people

00:04:40.456 --> 00:04:43.839
"and three chairs and I
might've been able just do this

00:04:43.839 --> 00:04:45.499
"even if I didn't do it systematically.

00:04:45.499 --> 00:04:49.439
"But what if I have many many more chairs?

00:04:49.439 --> 00:04:51.439
"What if I had 60 chairs?"

00:04:51.439 --> 00:04:53.919
Well, the number of scenarios
would get fairly large.

00:04:53.919 --> 00:04:58.059
What if I had five people and five chairs?

00:04:58.059 --> 00:04:59.519
Even this method right over here

00:04:59.519 --> 00:05:01.859
would take up a lot of paper space,

00:05:01.859 --> 00:05:03.099
or a lot of screen space.

00:05:03.099 --> 00:05:05.423
What do we do in that scenario?

00:05:06.038 --> 00:05:07.202
And the realization here is just

00:05:07.202 --> 00:05:09.708
thinking about what happened here.

00:05:10.707 --> 00:05:14.902
How many people could sit in seat one?

00:05:16.132 --> 00:05:17.333
If we're going seat by seat,

00:05:17.333 --> 00:05:18.913
how many possibilities were there

00:05:18.913 --> 00:05:20.453
to put in seat one?

00:05:20.453 --> 00:05:21.993
If you're seating in order,

00:05:21.993 --> 00:05:25.779
if you haven't filled
any of the seats yet,

00:05:27.948 --> 00:05:31.097
right at the beginning,
there's three possible people

00:05:31.097 --> 00:05:33.097
who could sit in seat one.

00:05:33.097 --> 00:05:35.101
You see it right over here, one

00:05:35.101 --> 00:05:38.726
two, three, because there's
three possible people.

00:05:39.541 --> 00:05:42.954
For each of those, how many
people could sit in seat two?

00:05:42.954 --> 00:05:45.414
For each of them, let's
say the scenario where A

00:05:45.414 --> 00:05:48.114
is sitting in seat one,
there's two possibilities

00:05:48.114 --> 00:05:50.464
for who could sit in seat two.

00:05:51.109 --> 00:05:52.728
Let me do this in a different color.

00:05:52.728 --> 00:05:54.968
So this, right over here.

00:05:54.968 --> 00:05:59.968
For each of these three,
there's two possibilities

00:06:00.728 --> 00:06:03.468
of who could sit in seat two.

00:06:03.468 --> 00:06:05.728
If A is sitting in seat
one, you could have

00:06:05.728 --> 00:06:07.488
either B or C in seat two.

00:06:07.488 --> 00:06:08.968
If B is sitting in seat one, you have

00:06:08.968 --> 00:06:10.868
either A or C for seat two.

00:06:10.868 --> 00:06:14.288
So for each of these three,
you have two possibilities

00:06:14.288 --> 00:06:16.532
of who could sit in seat two.

00:06:18.930 --> 00:06:21.024
So you have six possibilities
where you're filling in

00:06:21.024 --> 00:06:23.304
three times two, or you're filling in

00:06:23.304 --> 00:06:26.464
the scenarios for seats one and two.

00:06:26.464 --> 00:06:28.224
And then for each of those,

00:06:28.224 --> 00:06:33.224
how many scenarios are
who can sit in seat three?

00:06:34.330 --> 00:06:37.667
For each of those, there was only

00:06:37.667 --> 00:06:42.472
one possibility for who could sit

00:06:43.041 --> 00:06:46.544
in seat three.

00:06:46.544 --> 00:06:47.944
We see it right there, that's because

00:06:47.944 --> 00:06:50.684
there's only one person left
who hasn't sat down yet.

00:06:50.684 --> 00:06:52.784
So how many total possibilities are there?

00:06:52.784 --> 00:06:57.784
Well, three times two
times one is equal to six.

00:06:57.784 --> 00:07:00.383
And so if we were to do that
same exact thought exercise,

00:07:00.383 --> 00:07:03.262
with say, five seats, let's
go through that exercise.

00:07:03.262 --> 00:07:04.863
It's interesting.

00:07:04.863 --> 00:07:09.840
So if we have one, two, three, four, five.

00:07:09.840 --> 00:07:12.673
So this is five seats with
five people sitting down,

00:07:12.673 --> 00:07:14.373
and we want to figure out
how many scenarios are there

00:07:14.373 --> 00:07:17.189
for all the different people
and all the different seats.

00:07:17.189 --> 00:07:19.129
For the first seat, there's five

00:07:19.129 --> 00:07:21.569
different people who could sit in it.

00:07:21.569 --> 00:07:23.938
For each of those scenarios,
there's four people

00:07:23.938 --> 00:07:27.695
who haven't sat down yet who
could sit in the second seat.

00:07:27.695 --> 00:07:30.399
For each of those scenarios,
there's going to be

00:07:30.399 --> 00:07:32.179
three people who haven't sat down yet,

00:07:32.179 --> 00:07:33.869
so those are the people who
could sit in the third seat

00:07:33.869 --> 00:07:36.159
if we're filling out
the seats in this order.

00:07:36.159 --> 00:07:37.939
For each of all of these scenarios,

00:07:37.939 --> 00:07:40.979
there's two people left
who haven't sat down yet,

00:07:40.979 --> 00:07:42.919
who could sit in the fourth seat.

00:07:42.919 --> 00:07:44.659
And then for all of these scenarios,

00:07:44.659 --> 00:07:46.639
there's only one person
left who hasn't sat down

00:07:46.639 --> 00:07:48.826
who would have to sit in the fifth seat.

00:07:48.826 --> 00:07:50.784
If you had this exact
same thing, but instead of

00:07:50.784 --> 00:07:52.524
three chairs you had five chairs,

00:07:52.524 --> 00:07:54.024
the number of scenarios would be

00:07:54.024 --> 00:07:57.804
five times four times
three times two times one.

00:07:57.804 --> 00:08:00.864
Which is what, 20 times six,

00:08:00.864 --> 00:08:03.819
which is equal to 120 scenarios.

00:08:03.819 --> 00:08:07.894
And so you see that the
scenarios grow fairly quickly.

00:08:07.894 --> 00:08:10.028
And if you're wondering,
"Hey, this is kind of a neat

00:08:10.028 --> 00:08:12.268
mathematical thing. Three
times two times one."

00:08:12.268 --> 00:08:15.148
Or five times four times
three times two times one.

00:08:15.148 --> 00:08:17.128
If you start with a number
and you multiply that number

00:08:17.128 --> 00:08:20.028
times number one less than that

00:08:20.028 --> 00:08:22.388
all the way down to getting to one,

00:08:22.388 --> 00:08:25.618
that seems like neat kind of
fun mathematical operation.

00:08:25.618 --> 00:08:27.838
And lucky for you, or
maybe unlucky for you,

00:08:27.838 --> 00:08:29.318
because this might have
been your chance at fame,

00:08:29.318 --> 00:08:31.721
this operation has already been defined.

00:08:31.721 --> 00:08:35.663
It's called the factorial operation.

00:08:35.663 --> 00:08:37.159
So this thing right over here,

00:08:37.159 --> 00:08:41.195
this is the same thing as three factorial.

00:08:41.195 --> 00:08:42.936
You write this little exclamation mark.

00:08:42.936 --> 00:08:45.616
This right over here,
this is the same thing

00:08:45.616 --> 00:08:47.296
as five factorial.

00:08:47.296 --> 00:08:50.636
In general, if I said six factorial,

00:08:50.636 --> 00:08:53.076
that would be equal to six times five

00:08:53.076 --> 00:08:57.816
times four times three
times two times one.

00:08:57.816 --> 00:09:00.049
Hopefully you enjoyed that.