Exam-Style Questions on Combinatorics

Nine cats squeeze through a gap in a fence to get to a fish pond. There is only enough room for one cat at a time to squeeze through the gap.

Seefer and Sushi are two of the nine cats attracted to the fish pond.

Find the total number of possible ways in which the nine cats can squeeze through the gap if Seefer passes through

(a) in the position immediately after Sushi;

(b) in any position after Sushi.

Worked Solution

A fund-raising activity involves people choosing three balls from a jar containing balls numbered 1 to 30. A prize of £5 is awarded if the three numbers chosen sum to a multiple of three. It costs £2 to have a go.

(a) How many ways are there of choosing three numbers from the integers 1 to 30 that sum to a multiple of 3?

(b) What is the probability of person winning a prize?

(c) What is the expected profit made for the fund after 300 people have had a go?

Worked Solution

Fifty years ago there was a zoo where animals were housed in cages. There were eight animal cages in a line on the northern perimeter of the zoo. Only four cages were being used: one to hold the gorilla, one for the chimpanzees, one for the gibbons and one for the macaques.

(a) Find the number of ways the four cages can be assigned to the animals so that the cages are next to each other.

It was decided that the gorilla should have at least one empty cage between its cage and the next occupied cage. The other animal cages do not have to be next to each other.

(b) Find the number of ways the four cages can be assigned now.

Worked Solution