1
1 A committee of 5 people is to be formed from 6 doctors, 4 dentists and 3 nurses. Find the number of different
committees that could be formed if the committee contains all the nurses.
[1]
[Total: 1]
2 A committee of 5 people is to be formed from 6 doctors, 4 dentists and 3 nurses. Find the number of different
committees that could be formed if there are no restrictions.
[1]
[Total: 1]
3 360 different 4-digit numbers can be formed using the digits 1, 3, 4, 6, 7 and 9. Each digit is used once only
in any 4-digit number.
How many of these 4-digit numbers are even and greater than 6000?
[3]
[Total: 3]
2
4 There are 720 different 5-digit numbers that can be formed using the digits 1, 2, 3, 5, 7 and 8, if each digit
may be used only once in any number.
How many of these 5-digit numbers are even and greater than 30 000?
[4]
[Total: 4]
5 The number of combinations of n items taken 3 at a time is 6 times the number of combinations of n items
taken 2 at a time. Find the value of the constant n.
[4]
3
[Total: 4]
6 There are 6720 different 5-digit numbers that can be formed using five of the eight digits
1, 2, 3, 4, 5, 6, 7, 8 if each digit can be used once only.
Find how many of these 5-digit numbers are greater than 60 000.
[2]
[Total: 2]
7 The number of combinations of n items taken 3 at a time is 92n. Find the value of the constant n.
[4]
[Total: 4]
8 There are 360 different 4-digit numbers that can be formed using the digits 2, 3, 5, 7, 8 and 9, if each digit
may be used only once in any number.
How many of these numbers are divisible by 5?
[1]
[Total: 1]
4
9 There are 360 different 4-digit numbers that can be formed using the digits 2, 3, 5, 7, 8 and 9, if each digit
may be used only once in any number.
How many of these numbers are odd and greater than 7000?
[4]
[Total: 4]
10 Jack has won 7 trophies for sport and wants to arrange them on a shelf. He has 2 trophies for cricket, 4 trophies
for football and 1 trophy for swimming. Find the number of different arrangements if
(a) there are no restrictions,
[1]
5
(b) the football trophies are to be kept together,
[3]
(c) the football trophies are to be kept together and the cricket trophies are to be kept together.
[3]
[Total: 7]
6
11 A quiz team of 6 children is to be chosen from a class of 8 boys and 10 girls. Find the number of ways of
choosing the team if
(a) there are no restrictions,
[1]
7
(b) there are more boys than girls in the team.
[4]
[Total: 5]
12 Eight books are to be arranged on a shelf. There are 4 mathematics books, 3 geography books and 1 French
book.
8
(a) Find the number of different arrangements of the books if there are no restrictions.
[1]
(b) Find the number of different arrangements if the mathematics books have to be kept together.
[3]
9
(c) Find the number of different arrangements if the mathematics books have to be kept together and the
geography books have to be kept together.
[3]
[Total: 7]
13 A team of 6 players is to be chosen from 8 men and 4 women. Find the number of different ways this can be
done if
(a) there are no restrictions,
[1]
10
(b) there is at least one woman in the team.
[2]
[Total: 3]
14 A group of people is to be selected from 5 women and 3 men.
(a) Calculate the number of different groups of 4 people that have exactly 3 women.
[2]
11
(b) Calculate the number of different groups of at most 4 people where the number of women is the same
as the number of men.
[2]
[Total: 4]
15 Five different books are to be arranged on a shelf. There are 2 Mathematics books and 3 History books. Find
the number of different arrangements of books if
(a) the Mathematics books are next to each other,
[2]
(b) the Mathematics books are not next to each other.
[2]
[Total: 4]
12
16 A security code is to be chosen using 6 of the following:
• the letters A, B and C
• the numbers 2, 3 and 5
• the symbols * and $.
None of the above may be used more than once. Find the number of different security codes that may be
chosen if
(a) there are no restrictions,
[1]
(b) the security code starts with a letter and finishes with a symbol,
[2]
13
(c) the two symbols are next to each other in the security code.
[3]
[Total: 6]
17 A lock can be opened using only the number 4351. State whether this is a permutation or a combination of
digits, giving a reason for your answer.
[1]
[Total: 1]
18 There are twenty numbered balls in a bag. Two of the balls are numbered 0, six are numbered 1, five are
numbered 2 and seven are numbered 3, as shown in the table below.
Number on ball 0 1 2 3
Frequency 2 6 5 7
Four of these balls are chosen at random, without replacement. Calculate the number of ways this can be
done so that
(a) the four balls all have the same number,
[2]
14
(b) the four balls all have different numbers,
[2]
(c) the four balls have numbers that total 3.
[3]
[Total: 7]
19 (a) How many different 5-digit numbers can be formed using the digits 1, 2, 4, 5, 7 and 9 if no digit is
repeated?
[1]
19 (b) How many of these numbers are even?
[1]
(c) How many of these numbers are less than 60 000 and even?
[3]
[Total: 5]
15
20 (a) Find how many different numbers can be formed using 4 of the digits
1, 2, 3, 4, 5, 6 and 7 if no digit is repeated.
[1]
Find how many of these 4-digit numbers are
(b) odd,
[1]
(c) odd and less than 3000.
[3]
[Total: 5]
21 A 4-digit number is to be formed from the digits 1, 2, 5, 7, 8 and 9. Each digit may only be used once. Find
the number of different 4-digit numbers that can be formed if
16
(a) there are no restrictions,
[1]
(b) the 4-digit numbers are divisible by 5,
[2]
(c) the 4-digit numbers are divisible by 5 and are greater than 7000.
[2]
[Total: 5]
