I'm reading the following document. On page 2 we have the following formula:
$$\frac{N!}{(pN)!((1-p)N)!}\sim \frac{N^N}{(pN)^{pN}((1-p)N)^{((1-p)N)}}$$
It seems that $n!$ is being approximated with $n^n$, which is obviously false. However, I guess the following approximation works $\frac{m!}{n!}=\frac{m^m}{n^n}$. Any ideas why?
On
Recall Stirling's approximation $$ n!\approx n^ne^{-n}\sqrt{2\pi n}$$ so that we can approximate $\frac{n!}{m!k!}$ where $m+n=k$ as $$ \frac{n!}{m!k!}\approx\frac{n^ne^{-n}\sqrt{2\pi n}}{m^me^{-m}\sqrt{2\pi m}\cdot k^ke^{-k}\sqrt{2\pi k}}=\frac{n^n}{m^mk^k}\cdot \sqrt{\frac{n}{\pi mk}}$$ The error factor $\sqrt{\frac{n}{\pi mk}}$ can vary from $\approx \frac1{\sqrt\pi}$ when $m=1$ or $k=1$ to $\approx\sqrt{\frac{4}{\pi n}}$ when $m=k=\frac n2$.
Your approximation does not quite work, but the approximation in question should follow from Stirling's formula.