If are finite sets, then where each is the sum of the sizes of the intersections of each subset of sets from .
Proof:
Given an we want to show that contributes exactly to the sum. Suppose lies in exactly of the sets. Then contributes exactly to since that's the number of subsets of where each set contains (the intersection contains ). So to the total sum, contributes From the Binomial theoremBinomial theorem, this is simply except it's missing the term , so contributes exactly to the total sum.