TITLE: Growth rate of formula
QUESTION [5 upvotes]: I have formula: $\frac{(m+n)!}{m!n!}$
I am wondering what is growth rate of it. Can I say that it grows exponentially with m and n? Or maybe this is different growth rate?
Greetings,
Rnd

REPLY [2 votes]: This is probably a very close approximation for $n=m$:
$$
n!n!=(n!)^2 \approx \frac{n^2n}{e^2n} 2 \pi n\\
(2n)! \approx \frac{(2n)^{2n}}{e^{2n}}2 \sqrt{\pi n}
$$
Dividing $(2n)!$ through $n!^2$you get that 
$$
\frac{(2n)!}{n!^2} \approx \frac{4^n}{\sqrt{\pi n}}
$$