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0606/23 Oct/Nov 2025, Question 5: Permutations: Counting Arrangements

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

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The question, in short

From six distinct digits chosen from 1 to 9, count how many 6-digit numbers can be made under three different conditions.

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

Part (a) · 1 mark

Count all possible 6-digit numbers using six different digits from 1 to 9 with no restrictions.

  1. 1Arrange 6 of the 9 digits in order: ⁹P₆
  2. 2⁹P₆ = 9 × 8 × 7 × 6 × 5 × 4 = 60 480

Answer60 480

Part (b) · 2 marks

Count how many such 6-digit numbers exceed 700 000.

  1. 1The first digit must be 7, 8 or 9: 3 ways
  2. 2The other 5 digits from the remaining 8: ⁸P₅ = 6720
  3. 33 × 6720 = 20 160

Answer20 160

Part (c) · 3 marks

Count how many such 6-digit numbers exceed 750 000.

  1. 1Starts with 8 or 9: 2 × ⁸P₅ = 13 440
  2. 2Starts with 7 then 5, 6, 8 or 9: 4 × ⁷P₄ = 4 × 840 = 3360
  3. 3Total = 13 440 + 3360 = 16 800

Answer16 800

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