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How can we find the sum of the following series

$$\sum_{i=0}^p \binom{m-q+1+i}{i} \binom{n+q-1-i}{n-i}=\sum_{i=0}^p\frac{(m-q+1-i)!}{ i! (m-q+1)!}\frac{ ( n + q-1-i)!}{ (q-1)! (n-i)!}$$ where $p < n,m$?

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    Do you denote by $!n$ the factorial? Usually it is denoted by $n!$.2012-12-06
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    $!n$ is not the notation for factorial, but rather for [derangements](http://en.wikipedia.org/wiki/Derangement).2012-12-06

2 Answers 2