TITLE: maximize a function which contains factorials
QUESTION [6 upvotes]: Suppose I have a function 
$$ f(k) = \binom{500}{k} \binom{500}{1100-3k}$$
where $k$ is an integer from $200$ to $366$. How can I find the maximum analytically?

REPLY [5 votes]: Try $f(k)\ge f(k-1)$ and $f(k)\ge f(k+1)$.