I am interested in finding a lower bound of the sum: $$\sum_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right) \left(\genfrac{}{}{0pt}{}{m}{ki}\right)$$ 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 ChuVandermonde 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?
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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, 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. 

