Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that all 3 Es are together.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

1. [M1] 8! / 2! OR 7×8/k k! where k ∈ N, a! / 2(!) where a ∈ N
2. [A1] 20160
3. [B1] Total number of ways: 10! / (2!3!) (= 302400) (A) Accept unsimplified
4. [B1] With Ps together: 9! / 3! (= 60480) (B) Accept unsimplified
5. [M1] 10!/m - 9!/n, m, n integers or (A) – (B) if clearly identified
6. [A1] 241920
7. [M1] Number of ways Es at beginning and end = 8! / 2! (= 20160) divided by Total number of ways (from 7b) = 10! / (2!3!) (= 302400). OR (8!/(2!)) / (10!/(2!3!)) OR 1<k, l∈ N < 3, FT denominator from 7(b) or correct. OR 3/10 x 2/9
8. [A1] 1/15 or 0.0667
9. [M1] Cx seen alone, 1 ≤ x ≤ 4 (e.g. ⁵C₃, ⁵C₂, ⁵C₁, ⁵C₀)
10. [M1] Summing the number of ways for 3 or 4 correct scenarios (can be unsimplified), no incorrect scenarios. Scenarios: PEEE (⁵C₀=1), PEE__ (⁵C₁=5), PE___ (⁵C₂=10), P____ (⁵C₃=10)
11. [A1] Total = 26