```python
from sympy import symbols,Eq,Limit,E,oo,Function,factorial,binomial
n,x = symbols('n x')
Eq(Limit(E**(x/n),n,oo),Limit(E**(x/n),n,oo).doit())
```




$\displaystyle \lim_{n \to \infty} e^{\frac{x}{n}} = 1$




```python
P = Function('P') # probability function
r,N,f  = symbols('r N f')

N_choose_n = factorial(N)/(factorial(N-n)*factorial(n))
Eq(binomial(N, n),N_choose_n) # the number of ways of selecting an unordered set of n objects from a set of size N
```




$\displaystyle {\binom{N}{n}} = \frac{N!}{n! \left(N - n\right)!}$




```python
# Binomial Distribution
e1 = Eq(P(r,f & N) ,binomial(N, r)*f**r*(1-f)**(N-r))
e1
# probability of r given N and f 
```




$\displaystyle P{\left(r,N \wedge f \right)} = f^{r} \left(1 - f\right)^{N - r} {\binom{N}{r}}$




```python
E**((x+r)/factorial(r))
```




$\displaystyle e^{\frac{r + x}{r!}}$




```python
from sympy import E, factorial,Eq
E**factorial(E)
```




$\displaystyle e^{e!}$




```python
E**factorial(E).evalf()
```




$\displaystyle 70.8681052120441$




```python
Eq(E**factorial(E),E**factorial(E).evalf()) # issue raised at https://github.com/sympy/sympy/issues/22903

```




$\displaystyle \text{False}$




```python
# issue resolution 
Eq(E**factorial(E),E**factorial(E).evalf(), evaluate = False)
```




$\displaystyle e^{e!} = 70.8681052120441$



f14a624c72d6b9369e7067a9dc5e81e4881d5d07
ipynb
Jupyter Notebook
Personal_Projects/Binomial Distribution.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Personal_Projects/Binomial Distribution.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Personal_Projects/Binomial Distribution.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES