In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is selected from these 25 people to take part in a competition. For another competition, a team of 9 people consists of 2 swimmers, 3 cyclists and 4 runners. The team members stand in a line for a photograph.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [M1] ⁶Cₑ × ⁸Cբ × ¹¹Cᵍ, with e + f + g = 7 seen.
2. [B1] Correct outcome/value for 1 identified scenario, accept unsimplified, www.
3. [M1] Add values of 3 correct scenarios. No incorrect scenarios, no repeated scenarios. Condone ⁶Cₑ × ⁸Cբ × ¹¹Cᵍ, with e + f + g = 7 to identify S, C, R.
4. [A1] [Total =] 26964. cao
5. [M1] 2! × 3! × 4! × k, k an integer > 0. 1 can be implied.
6. [A1] =1728. If A0 scored SC B1 for 1728 www.
7. [M1] 6! × k, k an integer > 0. 1 can be implied.
8. [M1] (m! / (a! x b!)) × 7 × n × r; 6 < m ≤ 9; a = 1, 2; b = 1, 4; 1 ≤ n, r ≤ 6, n ≠ r.
9. [M1] (m! / (a! x b!)) × 7 × 6 × 5; 6 < m ≤ 9; a =1, 2; b = 1, 4.
10. [A1] 151 200. Condone 151 000. If A0 scored SC B1 for 151 200 www.
11. [M1] 6! × k, k an integer > 0. 1 can be implied.
12. [M1] (m! / (a! x b!)) × ⁷Pq, or (m! / (a! x b!)) × ⁷Cq × q! ; 6 ≤ m ≤ 9; a =1, 2; b = 1, 4; 1 ≤ q ≤ 6.
13. [M1] (m! / (a! x b!)) × ⁷P₃, or (m! / (a! x b!)) × ⁷C₃ × 3! ; 6 ≤ m ≤ 9; a =1, 2; b = 1, 4.
14. [A1] 151 200. Condone 151 000. If A0 scored SC B1 for 151 200 www.
15. [M1] 6! × k, k an integer > 0. 1 can be implied.
16. [M1] (m! / (a! x b!)) × 35 × q!; 6 < m < 9; a =1, 2; b = 1, 4; 1 ≤ q ≤ 3.
17. [M1] (m! / (a! x b!)) × 35 × 6; 6 < m < 9; a =1, 2; b = 1, 4.
18. [A1] 151 200. Condone 151 000. If A0 scored SC B1 for 151 200 www.
19. [M1] 9! – 7!r! – q, r an integer > 1, q an integer ≤ 0. 0 and 1 may be implied.
20. [M1] (s! / (a! x b! x c!)) - 7!3! - q; s = 8, 9; a =1, 2; b = 1, 3; c = 1, 4; q an integer ≥ 0.0 and 1 may be implied.
21. [M1] (s! / (a! x b! x c!)) - 7!3! - 3P2 × 6! × 6 × 7, 6 < s < 9, or (s! / (a! x b! x c!)) - 7!3! - 3! × 7! × 6, 6 < s < 9. a=1,2 b = 1,3 c = 1, 4.1 may be implied.
22. [A1] 151 200. Condone 151 000. If A0 scored SC B1 for 151 200 www.