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?

Choosing $k$ objects among $n$ objects is same as leaving $n-k$ objects among $n$ objects.

(Notice that there is no essential difference between the words "choose" and "leave".)

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Digression: I also consider this as one of the reasons why $0! = 1$ is a good definition.