Find the number of different arrangements of the 10 letters in the word CASABLANCA in which the two Cs are not together.

Mark scheme

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

1. [M1] 10!/(a!b!) - c, a ≠ b, a = 1, 2, b = 1, 4, with c being a positive integer.
2. [M1] e!/d!, e = 8, 9, 10, with d being a positive integer.
3. [A1] = 60480
4. [A1] Exact value only.
5. [A1] SC B1 for final answer 60480 www.
6. [M1] 8!/4! × f seen, with f being a positive integer.
7. [M1] g × 9×8/h, with g being a positive integer, h = 1, 2. g × 9C2 and g × 9P2 are acceptable.
8. [M1] 6!/2! × s, with s being a positive integer.
9. [M1] t!/r! × 4, r = 1, 2, 3 and t = 8, 7, 6.
10. [A1] 1440
11. [B1] Correct number of ways for identified scenarios of 2 or 3 As, accept unsimplified, www.
12. [M1] Add 3 values for 2, 3 and 4 As, no additional, incorrect or repeated scenarios. Accept unsimplified.
13. [A1] 25
14. [B1] Correct total number of ways for identified scenarios of 2 or 3 As, accept unsimplified, www (e.g., both values for AAC^^ and AA^^^ shown would be fine for 2As).
15. [M1] Add 6 values of appropriate scenarios only, no additional, incorrect or repeated scenarios. Accept unsimplified.