TITLE: Evaluate the following sum using a combinatorial argument
QUESTION [5 upvotes]: Evaluate the following sum using a combinatorial argument:
$$ \sum\limits_{k=0}^n {n \choose k} {m \choose k} $$
Can someone push me in the right direction with this? I thought for combinatorial proofs there has to be a left side and a right side where one side can be used to form a question? (if that makes sense? haha)
Is there a difference with combinatorial arguments? Any help would be greatly appreciated.

REPLY [3 votes]: As darij grinberg said in the comments, you’ll have to find the other side of the identity yourself; it’s a closed form, not a summation. 
HINT: You may find it easier to come up with a combinatorial interpretation of the summation if you rewrite it as 
$$\sum_{k=0}^n\binom{n}k\binom{m}{m-k}\;.$$