We have seen more than 10 problems based on
permutations and combinations. And we have used logic to solve all of those.
Once we understand the logic, there is no harm in using the formulae,
which we will see in this session.
We know that permutation is about arranging things.
And combination is about selecting things.
But we need to get a deeper understanding of this.
Is it extremely easy to figure out the difference between
arrangement and selection?
Before we get to a few examples, let's understand the notation.
If we have to arrange 'r' things out of 'n', it's written as 'npr'.
Arranging 'r' things out of 'n'.
We use 'p' as its permutations.
And for selecting 'r' things out of 'n' we write it as 'ncr', 'c' for combinations.
Let's zoom out and discuss each of these in detail.
Permutations is about arrangement .
But what does this arrangement tell us?
What is it that explains this word?
It tells us that the order matters.
The order in which the things are placed matters.
ABC and ACB will be different .
123 will be different from 231 and so on.
And what about combination?
Combination is selection!
Selection is simply picking a few things.
Here, the order does not matter.
Say from the digits 1, 2, 3, 4, 5 we picked up 1, 3 & 4.
Picking up 1, 3, 4 is the same as picking up 4, 1, 3.
They are not two different cases.
Let me give you a permutation example.
Assume we have to pick a team of three people out of the available ten.
One of them will be chosen as a vice president,
another as a manager and another as a product developer.
This is the classic case of permutation.
The order matters, if we select A, B and C.
Then it can be a case of VP, manager, product developer.
Or maybe A is chosen as a product developer,
B as vice president and C as a manager. Each case is different.
On the other hand if we are asked to select a leadership
team of three out of ten,
and told nothing else about it.
Then it is a case of combination.
The order does not matter.
Selecting A, B and C is the same as selecting B, C and A or C, A and B.
Take another example.
Say, 2 out of 5 people get ice cream.
One gets vanilla and the other gets chocolate.
This is the case of permutation as each of the two people selected,
may get either vanilla or chocolate flavoured ice cream.
And if we're just told that 2 out of 5 get ice cream,
it will be a case of combination.
We aren't told anything else!
Just that 2 out of 5 get ice cream.
So the order clearly does not matter.
Now that we have understood what permutations and combinations are
we can move on to the formulae.
First we look at 'npr'.
Arranging 'r' things out of 'n' distinct objects
and repetition is not allowed.
For example,
let's assume we have digits 1, 2, 3 & 4 .
And we have been asked the number of ways
in which we can form three-digit numbers using these four digits.
For this formula, we cannot consider cases such as
111 or
232 or
441.
The digits are repeated in each case.
So the objects have to be distinct and there can be no repetition.
So this equals 'n' factorial divided by 'n minus r' factorial.
Yes, that's the formula.
If we have to arrange three people out of ten.
It will equal 10 factorial divided by 10 minus 3 factorial.
Don't worry! We will look at numerical examples later!
For now, let's just look at the formulae .
We move on to combinations now!
'ncr'.
It's selecting 3 things from 10 distinct things.
This will equal 'n' factorial divided by 'r' factorial times 'n' minus 'r' factorial.
Yes, that's the formula for combinations.
If we have to select 2 out of 5,
the number of ways in which it can be done
is 5 factorial divided by 2 factorial
times 5 minus 2 factorial.
At first these formulae look scary.
But as you start practicing you will see that using these formulae .
gets us the answer really quick.
