I've asked similar question here Inequality for binomial coefficients, but with slightly different assumptions. I am curious what happend if $m, k$ are fixed.
Let $m \leq n, n \leq N$ and $0\leq k \leq m$.
I am wondering what is the dependence of $n$ and $N$ that for fixed $m, k$ $$ \frac{{N-m \choose n-k}}{{N \choose n}}\leq 1. $$
Thank you for your help.
