Let be subsets of a set that are not necessarily disjoint. Then the number of elements in the set is
The number of positive integers that are not divisible by any of the primes (§27.2(i)) is
With the notation of §26.15, the number of placements of nonattacking rooks on an chessboard that avoid the squares in a specified subset is
The number of ways of placing labeled objects into labeled boxes so that at least one object is in each box is
Note that this is also one of the counting problems for which a formula is given in Table 26.17.1. Elements of are labeled, elements of are labeled, and is onto.