Approximate this ratio of factorials

Question

I need to approximate the ratio $$ \frac{(2M)!}{(2M-N)!}\bigg/\frac{M!}{(M-N)!} $$

where $M$ and $N$ are huge, and $M\gg N$.

I tried taking the logarithm, applying Stirling's formula, and taking the exponential again but I didn't get a nice, simple order-of-magnitude answer.

Any suggestions?

Answer 1

The product of $N$ terms each about $2M$ divided by the product of $N$ terms each about $M$ is about $2^N$, and in fact slightly more since $\frac{2M-1}{M-1} \gt 2$ etc.

For example with $M=10^6$ and $N=10^3$, this suggests something slightly bigger than $2^{1000}$. The actual value is about $2^{1000.36}$, about $28\%$ higher than the approximation.

Answer 2

That looks like $2^N$: $$ \frac{(2M)!}{(2M-N)!}\bigg/\frac{M!}{(M-N)!}=\frac{2M·(2M-1)·(2M-2)·...·(2M-N+1)}{M·(M-1)·(M-2)·...·(M-N+1)}=\prod_{i=0}^{N-1}\left(2+\frac{i}{M-i}\right) $$ As M is very large, the fraction added to $2$ is very little, so the whole expression is aproximately $2^N$