Inclusion–Exclusion

Roughly speaking, a “sieve method” in enumerative combinatorics is a method for determining the cardinality of a set S that begins with a larger set and somehow subtracts off or cancels out the unwanted elements. Sieve methods have two basic variations: (1) We can first approximate our answer by an overcount, then subtract off an overcounted approximation to our original error, and so on, until after finitely many steps we have “converged” to the correct answer. This is the combinatorial essence of the Principle of Inclusion–Exclusion, to which this section and the next four are devoted. (2) The elements of the larger set can be weighted in a natural combinatorial way so that the unwanted elements cancel out, leaving only the original set S. We discuss this technique in Sections 2.5–2.7.

The Principle of Inclusion–Exclusion is one of the fundamental tools of enumerative combinatorics. Abstractly, the Principle of Inclusion–Exclusion amounts to nothing more than computing the inverse of a certain matrix. As such it is simply a minor result in linear algebra. The beauty of this principle lies not in the result itself, but rather in its wide applicability. We will give several examples of problems that can be solved by the Principle of Inclusion–Exclusion, some in a rather subtle way. First we state the principle in its purest form.