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?



Partial sums of the Chu--Vandermonde identity
Carla, I don't think it is a good idea to give a detailed solution here.
My point is that this problem is pretty standard: if you look inside
[N.G. de Bruijn's *Asymptotic Methods in Analysis*](http://books.google.com/books?isbn=0486642216),
especially in the 3rd chapter, you will find that your sum (generically) falls into the
category *c* (p. 54, a comparatively small number of terms somewhere in the middle);
the method is discussed in full details (Section 3.4) on the example of a very similar binomial sum.


