Permutation And Combination
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ADDITIONAL MATHEMATICS FORM 5 MODULE 11 PERMUTATIONS AND COMBINATIONS
1
11.0 Concept Map Multiplication Rules
The order of the objects in the chosen set is taken into consideration
The order of the objects in the chosen set is ____ taken into consideration
Permutation s
Combination s
If event A occurs in r ways and event B occurs in s ways The number of way is r xs
The number of combination of r objects taken from n different objects given
The number of permutation of n different objects is n! where n! = n( n-1)( n-2)…… The number of permutations of objects taken from n different object is given by
2
Permutations
11.1
A. PERMUTATIONS
11.1a Exercise A.1 1. Given the words PLACES. Find a. The number of ways of arranging all the
c. The number of ways arranging all the
alphabets in the given word.
alphabets in the given word so that the
Example Solution:
first alphabet is a vowel
6! = 6.5.4.3.2.1 = 720 b. The number of ways of arranging four of
d. Find the number of these
the alphabets in the given word so that last
arrangements in which the letters E and A
alphabet is S
are side by side.
Example Solution: The way to arrange alphabet S = 1 The way to arrange another 3 alphabets= 5 P 3 The number of arrangement = 1 x 5 P 3 =
60
2. How many four- digit numbers can be formed without repetitions from the digits 1, 2, 3, 4, 5, and 7 if that number. a) are even number
c) Consist of digit 5
b) are odd number
d) more than 4000
3
11.2
B. COMBINATIONS
11.2a Exercise B.1 1. How many ways to choose 5 books
2. In how many ways can committee of 3
from 20 different books
men and 3 women be chosen from a group
Example solution:
of 7 men and 6 women ? Example Solution:
The number of ways=
20
C5
The numbers of ways = 7 C 3 x 6 C 3
= 15504
= 700
2. A set of 20 students is made up of 10 students from each of two different year groups. Five students are to be selected from the set, and the order of selection is unimportant. Find a. the total number of possible selections. b. the number of selection in which there are at least two student from each of the two years- groups.
11.3
PAST YEAR SPM QUESTION 4
2003,Paper 1 1.Diagram below shows 5 letter and 3 digits. A
B
C
D
E
6
7
8
A code is to be formed using those letters and digits. The code must consist of 3 letters followed by 2 digits. How many codes can be formed if no letter or digit is repeated in each code ?
[ 3 marks]
2. A badminton team consist of 7 students. The team will be chosen from a group of 8 boys and 5 girls. Find the number of team that can be formed such that each team consists of i.
4 boys
ii.
Not more than 2 girl
[ 4 marks]
2004,Paper 1 3. Diagram below five cards of different letters. E
H
B
A
T
i.
Find the number of possible arrangements, in a row , of all the cards.
ii.
Find the number of these arrangements in which the letters E and A are side by side .
[ 4 marks]
2005,Paper 1 4. A debating team consists of 5 students. These 5 students are chosen from 4 monitors, 2 assistant monitors and 6 prefects. Calculate the number of different ways the team can be formed if i.
There is no restriction
ii.
The team contains only monitor and exactly 3 prefects
11.4 ASSESMENT 5
[4 marks]
1. Find the number of the arrangement of all nine letters of word SELECTION in which i.
The two letters E are next to each other
ii.
The two letters E are not next to each other
Solution:
2. Four girls and three boys are to be seated in a row. Calculate the number of possible arrangements i.
if all the three boys have to be seated together
ii.
a boy has to be seated at the centre
Solution:
6
3. Calculate the number of four digit even number can be formed from the digits 3, 4, 5, 6 and 9 without repetitions. Solution:
4. Three alphabets are chosen from the word WALID. Find the number of possible choice if i.
The alphabet A is chosen
ii.
The alphabet A and D is chosen
Solution:
5. A bowling team consists of 8 person. The team will be chosen from a group of 7 boys and 6 girls. Find the number of team that can be formed such that each team consists of i.
3 boys
ii.
Not more than 1 girl
Solution:
7
ANSWER Exercise A.1 1. c) 240
d) 240
2. a) 120
b) 240
c) 240
d) 180
Exercise B.1 2. a) 15504
b) 10800
Problem solving 1. i) 40320
ii) 141120
2. i) 720
ii) 2160
3. 72 4. i) 6
ii) 3
5 i) 210
ii) 6
Past year question 1. 360 2. i) 700
ii) 708
3. i) 120
ii) 48
4. i) 792
ii) 160
8
1
11.0 Concept Map Multiplication Rules
The order of the objects in the chosen set is taken into consideration
The order of the objects in the chosen set is ____ taken into consideration
Permutation s
Combination s
If event A occurs in r ways and event B occurs in s ways The number of way is r xs
The number of combination of r objects taken from n different objects given
The number of permutation of n different objects is n! where n! = n( n-1)( n-2)…… The number of permutations of objects taken from n different object is given by
2
Permutations
11.1
A. PERMUTATIONS
11.1a Exercise A.1 1. Given the words PLACES. Find a. The number of ways of arranging all the
c. The number of ways arranging all the
alphabets in the given word.
alphabets in the given word so that the
Example Solution:
first alphabet is a vowel
6! = 6.5.4.3.2.1 = 720 b. The number of ways of arranging four of
d. Find the number of these
the alphabets in the given word so that last
arrangements in which the letters E and A
alphabet is S
are side by side.
Example Solution: The way to arrange alphabet S = 1 The way to arrange another 3 alphabets= 5 P 3 The number of arrangement = 1 x 5 P 3 =
60
2. How many four- digit numbers can be formed without repetitions from the digits 1, 2, 3, 4, 5, and 7 if that number. a) are even number
c) Consist of digit 5
b) are odd number
d) more than 4000
3
11.2
B. COMBINATIONS
11.2a Exercise B.1 1. How many ways to choose 5 books
2. In how many ways can committee of 3
from 20 different books
men and 3 women be chosen from a group
Example solution:
of 7 men and 6 women ? Example Solution:
The number of ways=
20
C5
The numbers of ways = 7 C 3 x 6 C 3
= 15504
= 700
2. A set of 20 students is made up of 10 students from each of two different year groups. Five students are to be selected from the set, and the order of selection is unimportant. Find a. the total number of possible selections. b. the number of selection in which there are at least two student from each of the two years- groups.
11.3
PAST YEAR SPM QUESTION 4
2003,Paper 1 1.Diagram below shows 5 letter and 3 digits. A
B
C
D
E
6
7
8
A code is to be formed using those letters and digits. The code must consist of 3 letters followed by 2 digits. How many codes can be formed if no letter or digit is repeated in each code ?
[ 3 marks]
2. A badminton team consist of 7 students. The team will be chosen from a group of 8 boys and 5 girls. Find the number of team that can be formed such that each team consists of i.
4 boys
ii.
Not more than 2 girl
[ 4 marks]
2004,Paper 1 3. Diagram below five cards of different letters. E
H
B
A
T
i.
Find the number of possible arrangements, in a row , of all the cards.
ii.
Find the number of these arrangements in which the letters E and A are side by side .
[ 4 marks]
2005,Paper 1 4. A debating team consists of 5 students. These 5 students are chosen from 4 monitors, 2 assistant monitors and 6 prefects. Calculate the number of different ways the team can be formed if i.
There is no restriction
ii.
The team contains only monitor and exactly 3 prefects
11.4 ASSESMENT 5
[4 marks]
1. Find the number of the arrangement of all nine letters of word SELECTION in which i.
The two letters E are next to each other
ii.
The two letters E are not next to each other
Solution:
2. Four girls and three boys are to be seated in a row. Calculate the number of possible arrangements i.
if all the three boys have to be seated together
ii.
a boy has to be seated at the centre
Solution:
6
3. Calculate the number of four digit even number can be formed from the digits 3, 4, 5, 6 and 9 without repetitions. Solution:
4. Three alphabets are chosen from the word WALID. Find the number of possible choice if i.
The alphabet A is chosen
ii.
The alphabet A and D is chosen
Solution:
5. A bowling team consists of 8 person. The team will be chosen from a group of 7 boys and 6 girls. Find the number of team that can be formed such that each team consists of i.
3 boys
ii.
Not more than 1 girl
Solution:
7
ANSWER Exercise A.1 1. c) 240
d) 240
2. a) 120
b) 240
c) 240
d) 180
Exercise B.1 2. a) 15504
b) 10800
Problem solving 1. i) 40320
ii) 141120
2. i) 720
ii) 2160
3. 72 4. i) 6
ii) 3
5 i) 210
ii) 6
Past year question 1. 360 2. i) 700
ii) 708
3. i) 120
ii) 48
4. i) 792
ii) 160
8
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