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One time proof or similar to a string of these equalities
It is trivial to prove
$\sum_{k=0}^n\binom{n}k\binom{m}{k}=\binom{m+n}n$
but it is not trivial to prove a similar-looking sum $\sum_{k=0}^n\binom{m+k-1}k\binom{n+k-1}n$
We could assume $m=n$.
A:
HINT:
$\binom{m+n}{n}=\binom{m+n-1}n\implies \sum_{k=0}^n\binom{m+k-1}k\binom{n+k-1}n=\binom{m+n-1}n\sum_{k=0}^n\binom{m+k-1}k\binom{n+k-1}n$
Now expand the double sum and use
$\sum_{k=0}^n\binom{m+k-1}k=\binom{m+n}n$
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