Counting Principles
Multiplication principle, addition principle, inclusion-exclusion, and pigeonhole
Key Concepts
- Addition principle (disjoint cases) and multiplication principle (independent choices)
- Inclusion–exclusion handles overlapping sets
- The pigeonhole principle proves existence
- Counting the complement is often easier
Important Formulae
| Inclusion–exclusion | |A∪B| = |A| + |B| − |A∩B| |
| Pigeonhole | n items in k boxes ⇒ some box has ≥ ⌈n/k⌉ |
Quick Tips
- If direct counting is hard, count the complement and subtract.
- Use the pigeonhole principle whenever a problem asks to prove something must exist.
Sample Practice Questions
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The number of integers from 1 to 1000 divisible by 3 or 5 is:
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Answer: 467
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Burnside's lemma is used to count:
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Answer: Distinct objects under symmetry/rotation
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If 25 students each shake hands once with every other student, total handshakes =
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Answer: 300
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By the pigeonhole principle, if 13 items are placed in 12 boxes, at least one box has:
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Answer: At least 2 items
Practice Questions
Practise PRMO questions on Counting Principles. Answers are revealed after each question.
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What are the key concepts in Counting Principles for PRMO?
Addition principle (disjoint cases) and multiplication principle (independent choices); Inclusion–exclusion handles overlapping sets; The pigeonhole principle proves existence; Counting the complement is often easier.
Which formulae are important for Counting Principles in PRMO?
Inclusion–exclusion: |A∪B| = |A| + |B| − |A∩B|; Pigeonhole: n items in k boxes ⇒ some box has ≥ ⌈n/k⌉.
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