I am interested in finding a lower bound of the sum:
$$\sum_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right)
\left(\genfrac{}{}{0pt}{}{m}{k-i}\right)$$
when $d < k$ (and assuming both $n\geq k$, $m\geq k$). When $d=k$ this sum is equal to
$$\left(\genfrac{}{}{0pt}{}{n+m}{k}\right) $$ (the Chu--Vandermonde identity). What I would like to know if there is some good and standard lower bound for the first $d$ terms of this sum in terms of the relation between $d$ and $k$. Is there some standard reference book where I can hope to find such a bound?