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

Question

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?

Please elaborate.

Thanks

Answer 1

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

$$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}\sim \frac{(2n)^{2n}e^{-2n}\sqrt{4n\pi}}{n^{2n}e^{-2n}2n\pi}=\pi^{-1/2}2^{2n}n^{-1/2}$$