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Counting & Combinatorics

Pigeonhole principle, inclusion-exclusion, graph theory basics, and combinatorial arguments

Key Concepts
  • Bijections turn a hard count into an easier one
  • Invariants and monovariants prove impossibility or termination
  • The extremal principle: consider the largest or smallest case
  • Double counting evaluates a quantity two different ways
Important Formulae
Inclusion–exclusion |A∪B∪C| = Σ|A| − Σ|A∩B| + |A∩B∩C|
Handshake lemma Σ deg(v) = 2 × (number of edges)
Quick Tips
  • Look for an invariant (something that never changes) to prove a task is impossible.
  • The extremal principle — picking the maximal/minimal object — cracks many problems.
Sample Practice Questions

  1. In how many ways can 3 distinct books be arranged in a row?

    • 3
    • 6
    • 9
    • 27
    Show answer

    Answer: 6

  2. How many subsets does a set with 3 elements have?

    • 3
    • 6
    • 8
    • 9
    Show answer

    Answer: 8

  3. How many possible outcomes are there when a coin is flipped 3 times?

    • 3
    • 6
    • 8
    • 9
    Show answer

    Answer: 8

  4. If 5 people each shake hands once with every other person, how many handshakes occur?

    • 5
    • 10
    • 20
    • 25
    Show answer

    Answer: 10

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Practice Questions

Practise RMO questions on Counting & Combinatorics. Answers are revealed after each question.

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Combinatorics