Inclusion-exclusion theory

The inclusion-exclusion principle is a method used in probability and combinatorics to accurately count the number of items in multiple sets, especially when those sets overlap. It works by adding the sizes of individual sets, then subtracting the sizes of their overlaps to avoid double-counting. If there are overlaps among three or more sets, the process continues by alternately adding and subtracting the sizes of their intersections. This ensures we count each item exactly once, providing an exact total of elements in the combined sets.