📚 Counting Principles & Inclusion–Exclusion
Multiply stage counts with the fundamental counting principle or find the size of unions using inclusion–exclusion. Handy for license plate problems, scheduling, or Venn-diagram style questions.
1. Fundamental Counting Principle
Multiply the number of choices at each stage.
Total outcomes
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Switches to scientific notation for extremely large products.
2. Inclusion–Exclusion: Two Sets
|A ∪ B| = |A| + |B| − |A ∩ B|
Union size
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3. Inclusion–Exclusion: Three Sets
|A ∪ B ∪ C| = Σ|A| − Σ|A ∩ B| + |A ∩ B ∩ C|
Union size
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How to Use
- For the fundamental counting principle, list the number of choices per stage using commas, spaces, or an “×” separator, then click Multiply Stages.
- For inclusion–exclusion, enter the set sizes and intersection counts, ensuring intersections do not exceed the corresponding sets.
- Review warnings that highlight inconsistent or negative union totals so you can adjust inputs.
- Reset each card when switching between unrelated problems to keep calculations clear.
Formula References
- Brualdi, R. A. (2010). Introductory Combinatorics (5th ed.). Pearson.
- Lovász, L. (1979). Combinatorial Problems and Exercises. North-Holland.