I have the following identity: $$\sum_{k=0}^m \binom{m}{k}\binom{n}{r+k}=\binom{m+n}{m+r},$$ and my professor asked us to prove it in two different ways, I managed to do it in a combinatorial way (I know it as committee argument, thinking on a committee of $m+n$ persons and choosing $m+r$).
But for the second way of proving it, I have been struggling, I tried to express it as the factorial definition or use induction, but the algebra becomes too hard, also I tried to use some classic identities as Pascal or Vandermorde, but I don't find the way out. Is there something I am missing? There is another easy way to prove it?
Thanks.
