# Counting (CIE) Permutation and combination

1 (CIE 2012, s, paper 11, question 4)
(a) Arrangements containing 5 different letters from the word AMPLITUDE are to be made. Find
(i) the number of 5-letter arrangements if there are no restrictions,    [1]
(ii) the number of 5-letter arrangements which start with the letter A and end with the
letter E. [1]
(b) Tickets for a concert are given out randomly to a class containing 20 students. No student is given more than one ticket. There are 15 tickets.
(i) Find the number of ways in which this can be done. [1]

There are 12 boys and 8 girls in the class. Find the number of different ways in which
(ii) 10 boys and 5 girls get tickets, [3]
(iii) all the boys get tickets. [1]

2 (CIE 2012, s, paper 22, question 10)
(a) A team of 7 people is to be chosen from 5 women and 7 men. Calculate the number of different ways in which this can be done if
(i) there are no restrictions, [1]
(ii) the team is to contain more women than men. [3]
(b) (i) How many different 4-digit numbers, less than 5000, can be formed using 4 of the 6 digits 1, 2, 3, 4, 5 and 6 if no digit can be used more than once? [2]
(ii) How many of these 4-digit numbers are divisible by 5? [2]

3 (CIE 2012, w, paper 13, question 3)
A committee of 7 members is to be selected from 6 women and 9 men. Find the number of different committees that may be selected if
(i) there are no restrictions, [1]
(ii) the committee must consist of 2 women and 5 men, [2]
(iii) the committee must contain at least 1 woman. [3]

4 (CIE 2012, w, paper 21, question 9)
(a) An art gallery displays 10 paintings in a row. Of these paintings, 5 are by Picasso, 4 by Monet and 1 by Turner.
(i) Find the number of different ways the paintings can be displayed if there are no restrictions. [1]
(ii) Find the number of different ways the paintings can be displayed if the paintings by each of the artists are kept together. [3]

(b) A committee of 4 senior students and 2 junior students is to be selected from a group of 6 senior students and 5 junior students.
(i) Calculate the number of different committees which can be selected. [3]

One of the 6 senior students is a cousin of one of the 5 junior students.
(ii) Calculate the number of different committees which can be selected if at most one of
these cousins is included. [3]

5 (CIE 2012, w, paper 22, question 5)
A 4-digit number is formed by using four of the six digits 2, 3, 4, 5, 6 and 8; no digit may be used more than once in any number. How many different 4-digit numbers can be formed if
(i) there are no restrictions, [2]
(ii) the number is even and more than 6000? [3]

6 (CIE 2013, s, paper 11, question 3)
A committee of 6 members is to be selected from 5 men and 9 women. Find the number of different committees that could be selected if
(i) there are no restrictions, [1]
(ii) there are exactly 3 men and 3 women on the committee, [2]
(iii) there is at least 1 man on the committee. [3]

7 (CIE 2013, s, paper 12, question 2)
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
(i) there are no restrictions, [1]
(ii) the 4-digit numbers are divisible by 5, [2]
(iii) the 4-digit numbers are divisible by 5 and are greater than 7000. [2]

8 (CIE 2013, w, paper 11, question 7)
(a) (i) Find how many different 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8 and 9 if each digit may be used only once. [1]
(ii) Find how many of these 4-digit numbers are even. [1]

(b) A team of 6 people is to be selected from 8 men and 4 women. Find the number of different
teams that can be selected if
(i) there are no restrictions, [1]
(ii) the team contains all 4 women, [1]
(iii) the team contains at least 4 men. [3]

9 (CIE 2013, w, paper 23, question 2)
(i) 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
(ii) odd, [1]
(iii) odd and less than 3000. [3]

10 (CIE 2014, s, paper 11, question 10)
(a) How many even numbers less than 500 can be formed using the digits 1, 2, 3, 4 and 5? Each digit may be used only once in any number. [4]

(b) A committee of 8 people is to be chosen from 7 men and 5 women. Find the number of different committees that could be selected if
(i) the committee contains at least 3 men and at least 3 women, [4]
(ii) the oldest man or the oldest woman, but not both, must be included in the committee. [2]

11 (CIE 2014, s, paper 12, question 8)
(a) (i) 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]
(ii) How many of these numbers are even? [1]
(iii) How many of these numbers are less than 60 000 and even? [3]

(b) How many different groups of 6 children can be chosen from a class of 18 children if the class contains one set of twins who must not be separated? [3]

12 (CIE 2014, s, paper 13, question 7)
(a) A 5-character password is to be chosen from the letters A, B, C, D, E and the digits 4, 5, 6, 7. Each letter or digit may be used only once. Find the number of different passwords that can be chosen if
(i) there are no restrictions, [1]
(ii) the password contains 2 letters followed by 3 digits. [2]

(b) A school has 3 concert tickets to give out at random to a class of 18 boys and 15 girls. Find the number of ways in which this can be done if
(i) there are no restrictions, [1]
(ii) 2 of the tickets are given to boys and 1 ticket is given to a girl, [2]
(iii) at least 1 boy gets a ticket. [2]

13 (CIE 2014, w, paper 11, question 10)
(a) (i) Find how many different 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 and 6 if no digit is repeated. [1]
(ii) How many of the 4-digit numbers found in part (i) are greater than 6000? [1]
(iii) How many of the 4-digit numbers found in part (i) are greater than 6000 and are odd? [1]

(b) A quiz team of 10 players is to be chosen from a class of 8 boys and 12 girls.
(i) Find the number of different teams that can be chosen if the team has to have equal numbers
of girls and boys. [3]
(ii) Find the number of different teams that can be chosen if the team has to include the youngest and oldest boy and the youngest and oldest girl. [2]

14 (CIE 2014, w, paper 23, question 2)
A committee of four is to be selected from 7 men and 5 women. Find the number of different committees that could be selected if
(i) there are no restrictions, [1]
(ii) there must be two male and two female members. [2]

A brother and sister, Ken and Betty, are among the 7 men and 5 women.
(iii) Find how many different committees of four could be selected so that there are two male and two female members which must include either Ken or Betty but not both. [4]

1. (a)15120;210
(b)15504;3696;56
2. (a)792;196
(b)240;48
3. 6435;1890;6399
4. (a)3628800;17280
(b)150;110
5. (i)360 (ii)72
6. 3003;840;2919
7. 360;60;36
8. (a)360;120
(b)924;28;672
9. 840;480;140
10. (a)28(b)420,240
11. (a)720,240,144
(b)9828
12. (a)15120,480
(b)5456,2295,5001
13. (a)360;60;36
(b)44352;8008
14. 495;210;96