A group of 12 people consists of 3 boys, 4 girls and 5 adults.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.

1. M1 ⁵C₁ × ⁷C₄ or k, k integer ≥ 1 Condone P₁ for M1 only
2. A1
3. ⁵C₁ × ⁷C₄ = 5 × 35 = 175
4. M1 ³Cx × ⁴Cy × ⁵Cz, x + y + z = 5, x,y,z integers ≥1 Condone use of permutations for this mark
5. B1 2 appropriate identified outcomes correct, allow unsimplified
6. M1 Summing their values for 4 or 5 correct identified scenarios only (no repeats or additional scenarios), condone identification by unsimplified expressions
7. A1 Note: Only dependent upon M marks
8. Scenarios (Boys, Girls, Adults):
9. 2B, 1G, 2A: ³C₂ × ⁴C₁ × ⁵C₂ = 3 × 4 × 10 = 120
10. 2B, 2G, 1A: ³C₂ × ⁴C₂ × ⁵C₁ = 3 × 6 × 5 = 90
11. 2B, 3G, 0A: ³C₂ × ⁴C₃ × ⁵C₀ = 3 × 4 × 1 = 12
12. 3B, 1G, 1A: ³C₃ × ⁴C₁ × ⁵C₁ = 1 × 4 × 5 = 20
13. 3B, 2G, 0A: ³C₃ × ⁴C₂ × ⁵C₀ = 1 × 6 × 1 = 6
14. Total = 120 + 90 + 12 + 20 + 6 = 248
15. M1 8! × m, m an integer ≥ 1 Accept 8 × 7! for 8!
16. M1 3! × n, n an integer > 1
17. M1 p × ⁵P₂, p × ⁵C₂× 2, p × 20, p an integer > 1 If extra terms present, maximum 2/3 M marks available
18. A1 Exact value required
19. Ways to place 2 adults at ends = ⁵P₂ = 20
20. Treat 3 boys as a block (3!)
21. Remaining people: 1 block of boys, 4 girls, 3 adults. Total 8 items.
22. Arrangements of 8 items = 8!
23. Total arrangements = ⁵P₂ × 8! × 3! = 20 × 40320 × 6 = 4838400