$W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}p^n(1-p)^{N-n}$

W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}p^n(1-p)^{N-n}

$\displaystyle\frac{N!}{n!(N-n)!}\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}$

\displaystyle\frac{N!}{n!(N-n)!}\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}

$P(x)=\displaystyle\frac{1}{\sqrt{2\pi N^2p(1-p)}}e^{-(x-aNp)^2/2N^2p(1-p)}$

P(x)=1/(sqrt(2 * pi * N ^2 * p * (1 - p ))) exp(-( x - a * N * p )^2/(2* N ^2 * p * (1- p )))

$P(x)=\displaystyle\frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/2\sigma^2}$

P(x)=1/(sqrt(2 pi sigma^2))e^(-(x-mu)^2/2sigma^2)

$W_N(n) =\displaystyle\frac{ N !}{ n !( N - n )!} p ^ n (1- p )^{ N - n }$

W_N(n) =math.factorial( N )* p ^ n *(1- p )^( N - n )/(math.factorial( n )*math.factorial( N - n ))

$\sigma^2 = N ^2 p (1- p )$

sigma ^2 = N ^2 * p *(1- p )

$x=(n-Np)a$

x=(n-Np)a

$W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}p(1-p)}\left(1+\displaystyle\frac{x}{aNp}\right)^{-n-1/2}\left(1-\displaystyle\frac{x}{aN(1-p)}\right)^{-N+n-1/2}$

W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}p(1-p)}\left(1+\displaystyle\frac{x}{aNp}\right)^{-n-1/2}\left(1-\displaystyle\frac{x}{aN(1-p)}\right)^{-N+n-1/2}

$N!\sim\sqrt{2\pi N}\left(\displaystyle\frac{N}{e}\right)^N$

N!\sim\sqrt{2\pi N}\left(\displaystyle\frac{N}{e}\right)^N

$(N-n)!\sim\sqrt{2\pi(N-n)}\left(\displaystyle\frac{N-n}{e}\right)^{N-n}$

( N - n )!=sqrt(2* pi *( N - n ))*( N - n )/ e )^( N - n )

$n!\sim\sqrt{2\pi n}\left(\displaystyle\frac{n}{e}\right)^n$

n!\sim\sqrt{2\pi n}\left(\displaystyle\frac{n}{e}\right)^n

$\displaystyle\frac{n}{N}=p\left(1+\displaystyle\frac{x}{aNp}\right)$

\displaystyle\frac{n}{N}=p\left(1+\displaystyle\frac{x}{aNp}\right)

$\displaystyle\frac{N-n}{N}=(1-p)\left(1-\displaystyle\frac{x}{aN(1-p)}\right)$

\displaystyle\frac{N-n}{N}=(1-p)\left(1-\displaystyle\frac{x}{aN(1-p)}\right)

$\left(1+\displaystyle\frac{x}{aNp}\right)\sim e^{x/aNp-x^2/2a^2N^2p^2}$

\left(1+\displaystyle\frac{x}{aNp}\right)\sim e^{x/aNp-x^2/2a^2N^2p^2}

$\left(1-\displaystyle\frac{x}{aN(1-p)}\right)\sim e^{-x/aN(1-p)-x^2/2a^2N^2(1-p)^2}$

\left(1-\displaystyle\frac{x}{aN(1-p)}\right)\sim e^{-x/aN(1-p)-x^2/2a^2N^2(1-p)^2}

$\mu=aNp$

\mu=aNp

$u=\displaystyle\frac{x}{aNp}$

u=\displaystyle\frac{x}{aNp}

$u=\displaystyle\frac{x}{aN(1-p)}$

u=\displaystyle\frac{x}{aN(1-p)}