Find the condition where the product of two factorials is minimized I have to find the maximum value of a combination ${N \choose H} = \dfrac{N!}{H!(N-H)!}$ where $1 \leq H \leq N \leq 16$. My reasoning was to find the maximum value you have to maximize the numerator $N!$ and minimize the denominator $H!(N-H)!$. The maximum value of numerator occurs when $N = 16$. Then the term becomes, $\dfrac{16!}{H!(16-H)!}$. As, $1 \leq H \leq 16$, I can manually check for which $H$ value the term is maximized. But is there any other way to find out that the denominator is minimized when $H = 8$?
I tried to break now into three conditions, namely when $H > (16-H)$, $H < (16 - H)$ and $ H = 16 - H$, but I couldn't establish the relation between the three conditions. Any help would be appreaciated.

Hint:
Consider the ratio:
$$
\frac{\binom N{H+1}}{\binom NH}$$
and compare it with 1.
