Examples of Combinatorial Proof

Pascal's identity:
C(n+1,k) = C(n,k-1) + C(n,k)

The Binomial theorem:
(x+y)^n = sum^n_{j=0} C(n,j) x^{n-j}y^j

Cardinality of the power set of a finite set:
If S is a finite set with n elements, then S has 2^n subsets.