# StatMech: Binomial approximation

1. Sep 22, 2013

1. The problem statement, all variables and given/known data

The question requires me to approximate binomial distribution to get poisson distribution.
Show that N!/(N-n)!=N^n.

2. Relevant equations

N!/n!(N-n)! p^n q^(N-n)=Binomial distribution

3. The attempt at a solution

I expanded N!/(N-n)! and got: (N-1)(N-2)(N-3)....(N-n+2)(N-n+1). This didn't help me in getting the required approximation. So, then I wrote it as follows:( N-(n-(n-1)) ) ( N-( n- (n-2) ) )...( N-(n-2) ) ( N-(n-1) ).
It seem to have further complicated the question.
Thank you.

2. Sep 22, 2013

### UltrafastPED

Look at how the Stirling formula is derived ... it should be quite helpful.

3. Sep 22, 2013

I looked at the stirling formula derivation but I don't know how it is helpful here.
So I have solved it the other way.
[N-(n-(n-0))] [N-(n-(n-1))] [N-(n-(n-2))] [N-(n-(n-3))]......[N-( n-(3) )][N-( n-(2) )][N-( n-(1) )]
For N>>n, using this approximation once, I get n terms:
[N-n] [N-n] [N-n] ...... [N-n] [N-n] [N-n]
using the approximation again,
[N] [N] [N] ......[N] [N] [N] =N^n

My question is can I use this approximation selectively like I did in the above two steps.
Secondly, how was sterling formula derivation helpful? I used the x! formula and I get exponential(-n). Because n<<N, this term is big, making the entire answer zero.