Definition:
PIE is a combinatorial counting technique that calculates the size of the union of multiple sets by:
- Including individual set sizes
- Excluding pairwise intersections
- Including triple intersections
- And so on, alternating signs
Formula:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|Example:
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Students and Subjects Problem: In a class of 100 students:
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60 study Math (M)
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50 study Physics (P)
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40 study Chemistry (C)
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30 study both Math and Physics
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20 study both Math and Chemistry
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15 study both Physics and Chemistry
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10 study all three subjects
Find: How many students study at least one subject?
Solution using PIE:
|M ∪ P ∪ C| = |M| + |P| + |C| - |M ∩ P| - |M ∩ C| - |P ∩ C| + |M ∩ P ∩ C|
= 60 + 50 + 40 - 30 - 20 - 15 + 10
= 150 - 65 + 10
= 95Answer: 95 students study at least one subject.