Find the number of different arrangements of the 9 letters in the word ALLIGATOR in which the two As are together and the two Ls are together.

Mark scheme

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

1. [M1] 7! / (b!c!) b,c = 1,2
2. [A1] 5040 7! / (2!2!) oe, no further terms present.
3. Method 1 for first 3 marks: Arrangements of 6 letters including Ls between As
4. [M1] 5! × d, d integer > 1
5. [M1] e! x f x g, e = 5, 6, 7; f = 1, 5; g = 1, 2; f≠g, 1 can be implicit.
6. [A1] 1200
7. Method 2 for first 3 marks: Number of arrangements of LL^^^^^ – number of arrangements with the Ls split by an A
8. [M1] 6! x 2 – h h an integer 1 < h < 1440
9. [M1] k – 5! x 2 k an integer k > 240
10. [A1] 1200
11. Method 3 for first 3 marks: Alternative approaches to Method 1
12. [M1] LL treated as a single unit.
13. [B1] [Total number of arrangements =] 9! / (2!2!) = 90720 Accept unsimplified. May be seen as denominator of probability.
14. [B1 FT] their 1200 / their 90720 unsimplified B1 FT if their 1200 and their 90720 supported by work in this part. 0.0132
15. Method 1: Scenarios identified Both As and Ls removed
16. [B1] 1 correct, identified outcome/value for A, AL or AAL scenario, accept unsimplified ⁵C₅₋ₓ cannot be used in place of ⁵Cₓ
17. [M1] Add 4 values of appropriate scenarios, no incorrect scenarios, no repeated scenarios, accept unsimplified, condone use of permutations.
18. [A1] Value stated WWW.
19. Method 2: 1 A fixed, 1 L removed
20. [M1] ⁷C_h, 3 ≤ h ≤ 5
21. [B1] ⁷C₄ oe, no other terms, scenario identified.
22. [A1] Value stated.
23. Method 3: 1 A fixed, both Ls removed
24. [B1] Correct outcome/value for 1 identified scenario, accept unsimplified. WWW
25. [M1] Add 2 values of appropriate scenarios, no incorrect scenarios, no repeated scenarios, accept unsimplified, condone use of permutations.
26. [A1] Value stated.