This is Problem 14 from here.
Here's the meaning of combinatorial proof:
The statements in each problem are to be proved combinatorially, in most cases by exhibiting an explicit bijection between two sets. Try to give the most elegant proof possible. Avoid induction, recurrences, generating functions, etc., if at all possible.
And the problem is as follows:
Find a combinatorial proof of: $$ \sum_{i=0}^m{\binom{x+y+i}{i} \binom{y}{a-i} \binom{x}{b-i} }=\binom{x+a}{b} \binom{y+b}{a} $$ where $m=\min(a,b)$.
I have thought the problem for a long time but I failed to find the correct combinatorial construction. Can anyone help?
P.S.: Algebraic proof is also welcomed, since that I don't know how to prove it algebraically either. But it won't be accepted as a final answer to this problem.
EDIT: Algebraic proofs have been found here, and now only the combinatorial proof is missing.