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How can I simplify the following expression?

$$\sum_{j=0}^{k} \binom{n-j}{p} \binom{m+j}{q}$$

where $n,m,p,q,k$ are positive constants such that $n-k \ge p$ and $m \ge q$.

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    You'll need to tell us what $k$ is, too :)2012-12-03
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    As it stands, you cannot simplify it to a single expression.2012-12-03
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    I assume that $k=n-p$. In any case, try generating series.2012-12-03
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    Yeah, I updated the condition for k.2012-12-03
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    I tried to make few polynomial expansions to achieve the above form but all in vain.2012-12-03
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    If you take $k = n-p$ (i.e. maximal) then the expansion $$(1-x)^{-p-1} = \sum_{j=0}^{\infty} {p+j \choose p}x^j$$ (and a similar one for $q$) may help.2012-12-03
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    @WimC: If $n-k\le p$ and $m\le q$, then the sum is $\binom{n+m+1}{p+q+1}$, but that only applies to the question when $n-k=p$ and $m=q$.2012-12-03
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    @robjohn How you got that result if conditions are satisfied?2012-12-03
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    @Shashwat: I have added limited answer that covers that case.2012-12-04

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