0606/11 May/Jun 2025, Question 12: Combinations: Counting Selections
Cambridge IGCSE Additional Mathematics 0606 · Paper 11 (Non-calculator) · Non-calculator · 4 marks
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The question, in short
For n ≥ 6, use algebra to prove that ⁿC₅ − ⁿ⁻¹C₅ simplifies to ⁿ⁻¹C₄.
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Solution · 4 marks
- 1ⁿC₅ − ⁿ⁻¹C₅ = n!/((n − 5)!5!) − (n − 1)!/((n − 6)!5!)
- 2= (n − 1)!/((n − 6)!5!) × (n/(n − 5) − 1)
- 3= (n − 1)!/((n − 6)!5!) × 5/(n − 5)
- 4= (n − 1)!/((n − 5)! × 4!)
- 5= (n − 1)!/((n − 1 − 4)!4!) = ⁿ⁻¹C₄
AnswerⁿC₅ − ⁿ⁻¹C₅ = ⁿ⁻¹C₄ shown
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