Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. Five letters are selected from the 9 letters in the word DELIVERED. (c) Find the number of different selections if the 5 letters include at least one D and at least one E.

Mark scheme

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

1. [B1] 7!/2! - e, e a positive integer (including 0).
2. [M1] f-6!, f> 6!.
3. [M1] 7!/(a!b!) - 6!/(c!d!), a,c = 1,2 and b,d = 1,3.
4. [A1] 1800
5. [M1] 7! × s, s = positive integer > 1.
6. [M1] t! × 4 × 2, t = 8, 7, 6. Condone t! × 8.
7. [A1] 40320
8. [B1] [Total number of arrangements =] [9!=] 362880. Accept unsimplified. May be seen as denominator of probability.
9. [B1FT] Probability = 40320/362880 = 1/9. Accept unsimplified. their 40320 / their 362880. B1FT if their 40320 and their 362880 supported by work in this part. Condone their 362880 supported by calculation in 7(a).
10. [B1] 1 correct unsimplified outcome/value for one identified scenario excluding DDEEE. Note: 4C1 cannot be used for 4C3.
11. [M1] Add values of 6 appropriate scenarios, no additional, incorrect or repeated scenarios. Accept unsimplified.
12. [A1] [Total =] 99