A set of friends consists of 7 men and 4 women. Three of the men are brothers: Ali, Ben and Charlie.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [M1] 7! - k, k an integer ≥ 1.
2. [M1] m - 6! × n: n = 1, 2 where m an integer > 6! n = 1 can be implied.
3. [A1] = 3600
4. [M1] 5! × p where p an integer > 1.
5. [M1] q × 6 × 5, q × ⁶C₂, q × ⁶P₂, q × (6×5)/2, where q an integer > 1.
6. [A1] = 3600
7. [M1] 7! × r, where r is an integer, r > 1 or 4! × s, where s is an integer, s > 1.
8. [M1] 7! × 4! × t, where t is an integer, t ≥ 1, t = 1 can be implied.
9. [A1] 241920 CAO.
10. [M1] ¹¹C₆ × ⁵C₃ × [²C₂], u, v, w = 6, 3, 2 u ≠ v ≠ w.
11. [A1] = 4620
12. [B1] group of 6: ⁸C₃ × ⁵C₃ × (²C₂) = 560. (560 seen, accept un-simplified).
13. [B1] group of 3: ⁸C₆(× ²C₂ × ³C₃) = 28. (28 seen accept un-simplified).
14. [M1] Probability they are in same group = (their (560+28)) / their (c) or (their (560+28)) / 4620.
15. [A1] = 588/4620 = 7/55, 0.127.
16. [B1] Group of 6: 6/11 * 5/10 * 4/9 (= 4/33). (0.1212... seen, accept un-simplified).
17. [B1] Group of 3: 3/11 * 2/10 * 1/9 (= 1/165). (0.00606(06...) seen, accept un-simplified).
18. [M1] Probability they are in same group = their 4/33 + their 1/165.
19. [A1] = 7/55, 0.127.