TITLE: Combinatorial Proof of $\binom{n+m}{k} = \sum_{i=0}^k \binom{n}{i} \binom{m}{k-i}$
QUESTION [1 upvotes]: I want to prove the following combinatorially:
$$\binom{n+m}{k} = \sum_{i=0}^k \binom{n}{i} \binom{m}{k-i}$$
I know how to solve this algebraically but I need a solution based on ideas of combinatorics (like a background story and such). 
Any help would be appreciated.

REPLY [0 votes]: Not exactly combinatorial, but there are already three combinatorial answers.
What is the coefficient of $x^ky^{m+n-k}$ in $(x+y)^{m+n}$? And in $(x+y)^m(x+y)^n$?
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