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Maths Olympiad · topic 22 of 33 · free

Counting and arrangements — Maths Olympiad practice

Counting questions ask “in how many ways?”. When choices are made one after another, multiply them. When the order does not matter — a handshake, a match or a team of two — every pair has been counted twice, so divide by 2.

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How to solve it

Multiply the choices — “one choice after another means multiply the number of ways”

  1. Multiplication principle: a ways for the first choice and b ways for the next give a × b ways together.
  2. n different letters in a row: n × (n − 1) × … × 2 × 1 arrangements.
  3. Handshakes, matches or teams of 2 from n (order does not matter): n × (n − 1) ÷ 2. Captain and vice-captain (order matters): n × (n − 1).

Use it for: Outfits, routes, passwords, word arrangements, handshakes and tournament questions in Class 6–10.

Worked example: In how many different ways can the letters of the word KITE be arranged? → 24

  1. KITE has 4 different letters: 4 choices for the first place, then 3, then 2, then 1.
  2. 4 × 3 × 2 × 1 = 24.

Check: Words starting with K: 3 × 2 × 1 = 6, and each of the 4 letters can start, so 4 × 6 = 24 ✓

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✓ Written and checked by the TalentJR team · Updated 13 Oct 2026 · How we make our lessons

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