Principle of Inclusion-Exclusion
The Principle of Inclusion-Exclusion (abbreviated PIE) provides an organized method/formula to find the number of elements in the union of a given group of sets, the size of each set, and the size of all possible intersections among the sets.
We prove that each element is counted once.
Say that some element is in sets. Without loss of generality, these sets are
We proceed by induction. This is obvious for
If this is true for we prove this is true for For every set of sets not containing with size there is a set of sets containing with size In PIE, the sum of how many times these sets are counted is There is also one additional set of sets so is counted exactly once.
and so on.
2002 AIME I Problems/Problem 1 http://artofproblemsolving.com/wiki/index.php?title=2002_AIME_I_Problems/Problem_1#Problem
2011 AMC 8 Problems/Problem 6 https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8_Problems/Problem_6
2017 AMC 10B Problems/Problem 13 https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10B_Problems/Problem_13
2005 AMC 12A Problems/Problem 18 https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_18