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 all $m, k$ $\frac{{N-m \choose n-k}}{{N \choose n}}\leq 1.$
Thank you for your help.
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 all $m, k$ $\frac{{N-m \choose n-k}}{{N \choose n}}\leq 1.$
Thank you for your help.
If you try and force the numerator to be greater than the denominator which means making $N-m$ large relative to $n-k$, this happens at $m=k$. Here we have $\frac{\binom{N-k}{n-k}}{\binom{N}{n}} = \frac{\binom{N-k}{n}}{\binom{N}{n}}\;.$ This is clearly $ < 1$ and so for fixed $N,n$ this inequality is true for all $k,m$ given your criteria. In fact the inequality can be improved to $ <1 $ as mentioned above.