CONTENTS CONTENTS PAGES 11.0 CONCEPT MAP A. PERMUTATIONS a EXERCISE A B. COMBINATIONS a EXERCISE B PAST YEAR SPM

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1 PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM 5 MODULE 11 PERMUTATIONS AND COMBINATIONS 0

2 CONTENTS CONTENTS PAGES 11.0 CONCEPT MAP A. PERMUTATIONS a EXERCISE A B. COMBINATIONS a EXERCISE B PAST YEAR SPM QUESTION ASSESMENT 7 ANSWER 9 1

3 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 Permutations Combinations If event A occurs in r ways and event B occurs in s ways The number of way is r x s The number of combination of r objects taken from n different objects given n C r n! n r! r! 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 n n! P r n r! 2

4 11.1 A. PERMUTATIONS 11.1a Exercise A.1 1. Given the words PLACES. Find a. The number of ways of arranging all the alphabets in the given word. Example Solution: 6! = = 720 c. The number of ways arranging all the alphabets in the given word so that the first alphabet is a vowel b. The number of ways of arranging four of the alphabets in the given word so that last alphabet is S Example Solution: The way to arrange alphabet S = 1 d. Find the number of these arrangements in which the letters E and A are side by side. The way to arrange another 3 alphabets= 5 P 3 The number of arrangement = 1 x 5 P 3 = 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

5 11.2 B. COMBINATIONS 11.2a Exercise B.1 1. How many ways to choose 5 books from 20 different books Example solution: The number of ways= 20 C 5 = In how many ways can committee of 3 men and 3 women be chosen from a group of 7 men and 6 women? Example Solution: The numbers of ways = 7 C 3 x 6 C 3 = 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. 4

6 11.3 PAST YEAR SPM QUESTION 2003,Paper 1 1.Diagram below shows 5 letter and 3 digits. A B C D E 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] 5

7 2004,Paper 1 3. Diagram below five cards of different letters. H E 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 [4 marks] 6

8 11.4 ASSESMENT 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: 7

9 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: 8

10 ANSWER Exercise A.1 1. c) 240 d) a) 120 b) 240 c) 240 d) 180 Exercise B.1 2. a) b) Problem solving 1. i) ii) i) 720 ii) i) 6 ii) 3 5 i) 210 ii) 6 Past year question i) 700 ii) i) 120 ii) i) 792 ii) 160 9

6.1.1 The multiplication rule

6.1.1 The multiplication rule 1. There are 3 routes joining village A and village B and 4 routes joining village B and village C. Find the number of different ways of traveling from village A to village