# CIE Counting (-2020)

1. (CIE $0606 / 2018 / \mathrm{w} / 21 / \mathrm{q} 6)$

A 5 -digit code is to be formed from the digits $1,2,3,4,5,6,7,8,9$. Each digit can be used once only in any code. Find how many codes can be formed if

(i) the first digit of the code is 6 and the other four digits are odd,

(ii) each of the first three digits is even,

(iii) the first and last digits are prime.

2. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 22 / \mathrm{q} 6)$

(a) A 5-character code is to be formed from the 13 characters shown below, Each character may be used once only in any code. Letters $: \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}$ Numbers: $1,2,3,4,5,6,7.$ Find the number of different codes in which no two letters follow each other and no two numbers follow each other.

(b) A netball team of 7 players is to be chosen from 10 girls, 3 of these 10 girls are sisters. Find the number of different ways the team can be chosen if the team does not contain all 3 sisters.

3. (CIE $0606 / 2018 / \mathrm{w} / 23 / \mathrm{q} 7)$

A squad of 20 boys, which includes 2 sets of twins, is available for selection for a cricket team of II players. Calculate the number of different teams that can be selected if

(i) there are no restrictions,$[1]$

(ii) both sets of twins are selected,$[2]$

(iii) one set of twins is selected but neither twin from the other set is selected,$[2]$

4. (CIE $0606 / 2019 / \mathrm{w} / 11 / \mathrm{q} 11)$

(a) Jess wants to arrange 9 different books on a shelf. There are 4 mathematics books, 3 physics books and 2 chemistry books, Find the number of different possible arrangements of the books if

(i) there are no restrictions,$[1]$

(ii) a chemistry book is at each end of the shelf,$[2]$

(iii) all the mathematics books are kept together and all the physics books are kept together.

(b) 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

(i) there are no restrictions,[1]

(ii) there are more boys than girls in the team.$[4]$

5. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 6)$

(a) Eight books are to be arranged on a shelf. There are 4 mathematics books, 3 geography books and 1 French book.

(i) Find the number of different arrangements of the books if there are no restrictions.

(ii) Find the number of different arrangements if the mathematics books have to be kept together. [3]

(iii) 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]

(b) A team of 6 players is to be chosen from $8 \mathrm{men}$ and 4 women. Find the number of different ways this can be done if

(i) there are no restrictions,$[1]$

(ii) there is at least one woman in the team.$[2]$

6. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 13 / \mathrm{q} 7)$

(a) A 5-digit code is to be chosen from the digits $1,2,3,4,5,6,7,8$ and 9. Each digit may be used once only in any 5 -digit code. Find the number of different 5 -digit codes that may be chosen if

(i) there are no restrictions,$[1]$

(ii) the code is divisible by 5 ,$[1]$

(iii) the code is even and greater than 70000 .[3]

(b) A team of 6 people is to be chosen from 8 men and 6 women. Find the number of different teams that may be chosen if

(i) there are no restrictions,[1]

(ii) there are no women in the team,[1]

(iii) there are a husband and wife who must not be separated.[3]

7. (CIE $0606 / 2019 / \mathrm{s} / 13 / \mathrm{q} 9)$

(a) 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

(i) there are no restrictions,$[1]$

(ii) the football trophies are to be kept together,$[3]$

(iii) the football trophies are to be kept together and the cricket trophies are to be kept together.$[3]$

(b) A team of 8 players is to be chosen from 6 girls and 8 boys. Find the number of different ways the team may be chosen if

(i) there are no restrictions,$[1]^{2}$

(ii) all the girls are in the team,$[1]$

(iii) at least 1 girl is in the team.

8. (CIE $0606 / 2019 / \mathrm{s} / 21 / \mathrm{q} 9)$

(a) Eleven different television sets are to be displayed in a line in a large shop.

(i) Find the number of different ways the televisions can be arranged. Of these television sets, 6 are made by company $A$ and 5 are made by company $B$.

(ii) Find the number of different ways the televisions can be arranged so that no two sets made by. company $A$ are next to each other.

(b) A group of people is to be selected from 5 women and 3 men.

(i) Calculate the number of different groups of 4 people that have exactly 3 women.

(ii) Calculate the number of different groups of at most 4 people where the number of women is the same as the number of men.

9. (CIE $0606 / 2019 / \mathrm{m} / 22 / \mathrm{q} 1)$

A band can play 25 different pieces of music. From these pieces of music, 8 are to be selected for a concert.

(i) Find the number of different ways this can be done.

The 8 pieces of music are then arranged in order.

(ii) Find the number of different arrangements possible.

The band has 15 members, Three members are chosen at random to be the treasurer, secretary and agent.

(iii) Find the number of ways in which this can be done.[1]

10. (CIE $0606 / 2019 / \mathrm{w} / 22 / \mathrm{q} 3)$

A 5 -digit code is formed using the following characters.

No character can be repeated in a code. Find the number of possible codes if

(i) there are no restrictions,$[2]$

(ii) the code starts with a symbol followed by two letters and then two numbers,$[2]$

(iii) the first two characters are numbers, and no other numbers appear in the code.

11. (CIE $0606 / 2020 / \mathrm{w} / 11 / \mathrm{q} 5)$

(a) (i) Find how many different 4 -digit numbers can be formed using the digits $1,3,4,6,7$ and $9 .$

Each digit may be used once only in any 4 -digit number.$[1]$

(ii) How many of these 4 -digit numbers are even and greater than $6000 ?$$[3] (b) A committee of 5 people is to be formed from 6 doctors, 4 dentists and 3 nurses. Find the number of different committees that could be formed if (i) there are no restrictions,[1] (ii) the committee contains at least one doctor, (iii) the committee contains all the nurses. 12. (\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 12 / \mathrm{q} 4) (a) (i) Find how many different 5 -digit numbers can be formed using the digits 1,2,3,5,7 and 8 , if each digit may be used only once in any number. (ii) How many of the numbers found in part (i) are not divisible by 5 ?[4] (b) The number of combinations of n items taken 3 at a time is 6 times the number of combinations. of n items taken 2 at a time. Find the value of the constant n 13. (CIE 0606 / 2020 / \mathrm{w} / 12 / \mathrm{q} 8) (a) Find the number of ways in which 12 people can be put into 3 groups containing 3,4 and 5 people respectively. (b) 4 -digit numbers are to be formed using four of the digits 2,3,7,8 and 9 . Each digit may be used once only in any 4 -digit number. Find how many 4 -digit numbers can be formed if (i) there are no restrictions,[1] (ii) the number is even,[1] (iii) the number is greater than 7000 and odd.[3] 14. (CIE 0606 / 2020 / \mathrm{m} / 12 / \mathrm{q} 9) (a) (i) Find how many different 4 -digit numbers can be formed using the digits 2,3,5,7,8 and 9 , if each digit may be used only once in any number.[1] (ii) How many of the numbers found in part (i) are divisible by 5 ?$$[1]$

(iii) How many of the numbers found in part (i) are odd and greater than $7000 ?$

(b) The number of combinations of $n$ items taken 3 at a time is $92 n$. Find the value of the constant $n$.$[4]$

15. $(\mathrm{CIF} 0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 4)$

(a) In an examination, candidates must select 2 questions from the 5 questions in section $\mathrm{A}$ and select 4 questions from the 8 questions in section B. Find the number of ways in which this can be done.

(b) The digits of the number 6378129 are to be arranged so that the resulting 7 -digit number is even. Find the number of ways in which this can be done.

16. (CIE 0606/2020/s/23/q4)

(a) (i) Find how many different 5 -digit numbers 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.

(ii) Find how many of these 5 -digit numbers are greater than 60000 .

(b) A team of 3 people is to be selected from 4 men and 5 women. Find the number of different teams that could be selected which include at least 2 women.

17. (CIE 0606/2020/w/23/q6)

A 4 -digit code is to be formed using 4 different numbers selected from $1,2,3,4,5,6,7,8$ and $9 .$ Find how many different codes can be formed if

(a) there are no restrictions,$[1]$

(b) only prime numbers are used,$[1]$

(c) two even numbers are followed by two odd numbers,$[2]$

(d) the code forms an even number.$[2]$

1. (i) 120

(ii) 720

(iii) 2520

2. (a) 11340

(b) 85

3. (i) 167960 (ii) 11440

(iii) 22880 (iv) 45760

4. (a) (i) 362880 (ii) 10080

(iii) 3456

(b) (i) 18564 (ii) 3738

5. (a) (i) 40320

(ii) 2880

(iii) 864

(b) (i) 924(ii) 896

6. (a) (i) 15120 (ii) 1680 (iiii) 2310

(b) (i) 3003 (ii) 28 (iii) 1419

7. (a) (i) $5040($ ii $) 576$ (iii) 288

(b) (i) 3003 (ii) 28 (iii) 3002

8. (a) (i) 39916800

(ii) 86400

(b) (i) 30 (ii) 45

9. (i) 1081575

(ii) 40320

(iii) 2730

10. (i) 240240

(ii) 1800 (iii) 10080

11. (a) (i) 360 (ii) 60

(b) (i) 1287

(ii) 1266 (iii) 451234556789910

12. (a) (i) 720 (ii) 600

(iii) 168 (b) $n=20$

13. (a) 27720 (b) (i) 120

(ii) 48 (iii) 42

14. (a) (i) $360(\mathrm{i}$ i) 60 (iii) 120

(b) $n=25$

15. (a) 700 (b) 2160

16. (a) (i) 6720

(ii) 2520  (b) 50

17. (a) 3024  (b) 24

(c) 240 (b) 1344