November 2025 (9709/52). Question 7

(a) Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which the three Os are together and the two Ls are not next to each other [4 marks]

(b) Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which there are exactly 5 letters between the two Ls. [3 marks]

Two letters are chosen at random from the 10 letters in the word ZOOLOGICAL.

(c) Find the probability that these two letters are different. [3 marks]

9709/51/M/J/25q2 (Mostly Probability, but requires Combinatorics. Also easier with Geometric Distributions – later chapter) – Mark Scheme

(a) Find the number of different arrangements of the 8 letters in the word KANGAROO in which the two As are together and the two Os are not together. [3 marks]

A fair 8-sided dice has faces labelled K, A, N, G, A, R, O, O. The dice is rolled repeatedly.

(b) Find the probability that fewer than 6 rolls of this dice are required to obtain an A. [2 marks]

(c) Find the probability that the second A is obtained on the 6th roll of the dice. [2 marks]

9709/51/M/J/25q5 – Mark Scheme

In a group of 20 musicians, there are 9 guitarists, 6 pianists and 5 drummers.

6 musicians are selected from these 20 to perform at a concert.

(a) Find the number of different ways in which the 6 musicians can be selected if there must be at least 3 guitarists, at most 2 pianists and exactly 1 drummer. [4 marks]

Three bands will be selected from the original group of 20 musicians. Each band will consist of 3 guitarists, 1 pianist and 1 drummer. No musician can be in more than one band. The first band selected will play a concert in France, the second band selected will play in Italy and the third band selected will play in Spain.

(b) Find the number of different ways in which these three bands can be selected. [3 marks]

9709/52/M/J/25q6 – Mark Scheme

(a) Find the number of different ways in which the 10 letters in the word AMALGAMATE can be arranged so that there is an M at the beginning, an M at the end and no As are together. [3 marks]

(b) Find the number of different ways in which the 10 letters in the word AMALGAMATE can be arranged with exactly 3 letters between the two Ms. [3 marks]

Five letters are selected from the 10 letters in the word AMALGAMATE.

(c) Find the number of different selections in which the five letters include at least one M and at least two As. [3 marks]

9709/53/M/J/25q7 – Mark Scheme

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

(a) Find the number of different arrangements of the 7 men in a line in which Ali and Ben do not stand next to each other. [3 marks]

(b) Find the number of different arrangements of the 7 mean and 4 women in a line in which all the men stand together and all the women stand together. [3 marks]

(c) In how many ways can the 7 men and 4 women be divided into a group of 6, a group of 3 and a group of 2 if there are no restrictions? [2 marks]

(d) The 7 men and 4 women are divided at random into a group of 6, a group of 3 and a group of 2.

Find the probability that Ali, Ben and Charlie are all in the same group. [4 marks]