One of our very first counting principles was the sum principle which says that the size of a union of disjoint sets is the sum of their sizes. Computing the size of overlapping sets requires, quite naturally, information about how they overlap. Taking such information into account will allow us to develop a powerful extension of the sum principle known as the “principle of inclusion and exclusion.”
Thumbnail: Inclusion–exclusion illustrated by a Venn diagram for three sets. Image used with permission (CC BY-SA 3.0; Wikipedia).