The combinations (n choose k) tell you how many ways there are to take k things from n, and so count that subset of the power set. And since the sets of k thin

The combinations (n choose k) tell you how many ways there are to take k things from n, and so count that subset of the power set. And since the sets of k things are obviously distinct from the sets of j != k things, there is no overlap. This gives the well-known result that 2^n = Sum n choose k.