# Complexity analysis of m!/n!(m-n)!

Given the runtime of an algorithm to be m!/(n!*(m-n)!) That is mCn, where both m and n are variables, is the complexity factorial or polynomial? Or is it something else?

The growths could be linear or constant depending on the relationship between $$m$$ and $$n$$ as they grow large. For example, if $$n$$ grows with $$m$$ as in $$m=n+1$$, then $$\binom{m}{n}=\binom{n+1}{n}=n+1$$.
Now, if you are interested in the maximum growth, then the binomial coefficients $$\binom{m}{n}$$ are largest when $$n$$ is half of $$m$$. Put $$m=2n$$ and use Stirling's approximation for the factorial. You get
$$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}\sim \frac{(2n)^{2n}e^{-2n}\sqrt{4n\pi}}{n^{2n}e^{-2n}2n\pi}=\pi^{-1/2}2^{2n}n^{-1/2}$$