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0606/23 May/Jun 2025, Question 6: Permutations: Counting Arrangements

Cambridge IGCSE Additional Mathematics 0606 · Paper 23 (Calculator) · Calculator · 6 marks

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The question, in short

Count arrangements for three separate combinatorics situations: forming a 5-digit number above 50 000 from five of the digits 1 to 9, arranging 13 people with no two red-haired adjacent, and choosing 5 people with at least one brown-haired.

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Part (a) · 2 marks

Using five different digits from 1 to 9, count how many 5-digit numbers exceed 50 000.

  1. 1First digit must be 5, 6, 7, 8 or 9: 5 choices
  2. 2Remaining four digits: 8 × 7 × 6 × 5
  3. 35 × 8 × 7 × 6 × 5 = 8400

Answer8400

Part (bi) · 2 marks

Arrange 7 red-haired and 6 brown-haired people in a line so that no two red-haired people are adjacent; find the number of arrangements.

  1. 1With 7 red and 6 brown, the only pattern is R B R B R B R B R B R B R
  2. 2Arrange the red-haired people: 7! ways
  3. 3Arrange the brown-haired people: 6! ways
  4. 47! × 6! = 3 628 800

Answer3 628 800

Part (bii) · 2 marks

From 7 red-haired and 6 brown-haired people, count the selections of 5 that include at least 1 brown-haired person.

  1. 1All selections: ¹³C₅ = 1287
  2. 2Selections with no brown hair: ⁷C₅ = 21
  3. 31287 − 21 = 1266

Answer1266

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Also searched as: 0606/23/M/J/25 · 0606_s25_qp_23 · 0606 s25 23 · question 6.
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