Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 4 / 6E / ii / Solution

For finitely many finite sets $A_1,\dots ,A_k$, the inclusion-exclusion principle statesTo see why, an element belonging to exactly $t\ge 1$ of the sets contributesby the binomial theorem. Elements outside the union contribute zero. Equivalently, if these sets lie in a finite universe $U$, the number avoiding all of them is