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0606/21 May/Jun 2025, Question 3: Combinations: Counting Selections

Cambridge IGCSE Additional Mathematics 0606 · Paper 21 (Calculator) · Calculator · 5 marks

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The question, in short

Sports club has 5 runners, 4 swimmers, 3 gymnasts. Find arrangements in a line with each sport's members together, and selections of 4 including at least one of each sport.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Find the number of straight-line arrangements of all 12 members when the runners, swimmers and gymnasts each stand as a single block.

  1. 1Arrange the 3 groups: 3! = 6 ways
  2. 2Within groups: 5! × 4! × 3! = 120 × 24 × 6
  3. 3Total = 3! × 5! × 4! × 3! = 103 680

Answer103 680

Part (b) · 3 marks

Find the number of ways to choose 4 members so that at least one runner, one swimmer and one gymnast are included.

  1. 1One type must provide 2 members, the others 1 each
  2. 22 runners: ⁵C₂ × ⁴C₁ × ³C₁ = 10 × 4 × 3 = 120
  3. 32 swimmers: ⁵C₁ × ⁴C₂ × ³C₁ = 5 × 6 × 3 = 90
  4. 42 gymnasts: ⁵C₁ × ⁴C₁ × ³C₂ = 5 × 4 × 3 = 60
  5. 5Total = 120 + 90 + 60 = 270

Answer270

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