CAIE A-Level Mathematics AS 5.2 Permutations and Combinations Questions

Practise counting arrangements and selections with repeated objects, fixed positions, adjacency restrictions and case conditions by choosing permutations or combinations correctly.

Exam points

divide factorial arrangements by repeated-object factorials before applying position restrictions
use combinations for unordered selections and split minimum or maximum conditions into valid cases
count the unrestricted set and subtract arrangements that violate adjacency or grouping rules

Covered syllabus points

5.2.1—Permutations and combinations 5.2.2—Arrangement restrictions

[Maximum number: 10]

A new village social club has 10 members of whom 6 are men and 4 are women. The club committee will consist of 5 members.

(a)

In how many ways can the committee of 5 members be chosen if it must include at least 2 men and at least 1 woman?

[ 4 ]

The valid committee compositions are:
$\binom64\binom41+\binom63\binom42+\binom62\binom43 =60+120+60=240.$

(b)

The 10 members of the club stand in a line for a photograph.

How many different arrangements are there of the 10 members if all the men stand together and all the women stand together?

[ 2 ]

Arrange the six men as a block and the four women as a block. The two blocks can be ordered in either way:
$6!\times4!\times2=34560.$

(c)

For a second photograph, the members stand in two rows, with 6 on the back row and 4 on the front row. Olly and his sister Petra are two of the members of the club. How many different arrangements are there of the 10 members in which Olly and Petra stand next to each other on the front row?

[ 4 ]

Olly and Petra must be in the front row. Arrange them in 2! orders, choose and arrange two of the remaining eight people in the other front-row positions, and arrange the remaining six in the back row:
$\binom82\times3!\times2!\times6!=241920.$

[Maximum number: 4]

(a)

How many different arrangements are there of the 9 letters in the word RECORDERS?

[ 1 ]

The repeated letters are the three R's and the two E's. Hence the number of arrangements is
$\frac{9!}{2!\,3!}=30240.$

(b)

How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together?

[ 3 ]

With an E at each end, the number of arrangements is 7!/3!. If the three R's are together, the number is 5!. Therefore,
$\frac{7!}{3!}-5!=840-120=720.$

[Maximum number: 10]

Figure for Question 3 — CAIE A-Level Mathematics AS

In a restaurant, the tables are rectangular. Each table seats four people: two along each of the longer sides of the table (see diagram). Eight friends have booked two tables, X and Y. Rajid, Sue and Tan are three of these friends.

When the friends arrive at the restaurant, Rajid and Sue now decide to sit at table X on the same side as each other. Tan decides that he does not mind at which table he sits.

(a)

The eight friends will be divided into two groups of 4, one group for table X and one group for table Y.

Find the number of ways in which this can be done if Rajid and Sue must sit at the same table as each other and Tan must sit at the other table.

[ 3 ]

Choose two of the five friend-and-wife pairs for the first table and assign the remaining pairs to the second table:
$\binom52\times2=20.$

(b)

Find the number of different seating arrangements for the 8 friends.

[ 3 ]

Arrange the six people in one seating order, choose the two table orientations, and account for the two table assignments:
$\binom62\times2\times2\times2\times4!=2880.$

(c)

As they leave the restaurant, the 8 friends stand in a line for a photograph.

Find the number of different arrangements if Rajid and Sue stand next to each other, but neither is at an end of the line.

[ 4 ]

Treat Rajid and Sue as a block, subtract arrangements in which the four relevant people are together, or equivalently count directly. The number of arrangements is
$7!\times2-6!\times4=7200.$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question (c)