Combinatorial interpretation of the identity: ${n \choose k} = {n \choose n-k}$ What is the combinatorial interpretation of the identity: ${n \choose k} = {n \choose n-k}$?
Proving this algebraically is trivial, but what exactly is the "symmetry" here. Could someone give me some sort of example to help my understanding? 
EDIT: Can someone present a combinatorial proof?

Let's say you have 3 apples and you want to choose one of them.  Then you are left with an apple, which is the same, in a sense, as choosing that apple. 
