6.1.1 The multiplication rule
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1 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 C via village B. 2. There are 2 bus companies that provide transport from town E to town F and 3 bus companies that provide transport from town F to town G. Find the number of ways a person can travel from town E via to town G via town F by taking a bus. 3. There are 3 boys and 2 girls in a group. Find the number of ways to pick a boy and a girl for a Mathematics quiz. 4. Lim has to choose a shirt and a trouser from 6 shirts and 4 trousers in his cupboard. Find the possible number of ways he can do this. [6] 5. Team A which consists of 5 players wishes to have badminton matches with team B which consists of 6 players. Find the number of different single matches that can be held. [6] 6. A B C D [24] The diagram above shows cards of a game. If a player is going to select a letter card and a digit card, find the number of different ways can this be done. [30] 7. Hashim can choose to go to office by car or by bus. Find the number of different ways Hashim can go to school for three consecutive working days. 8. A football team plays four friendly matches in a month. Find the number of possible outcomes from the four matches. 9. There are 5 Malays, 3 Chinese and 2 Indians in a group. Find the number of ways can a Malay, a Chinese and an Indian be chosen from the group. [8] [16] 10.There are 2 pieces of cakes on plate A, 3 pieces of cakes plate B and 4 pieces of cakes on plate C. If a boy is going to select a cake from each plate, find the number of different ways can this be done. [30] [24] Permutations & Combinations 1
2 6.1.2 Permutations of n different objects. 1. In how many different ways can 5 different books be arranged on a shelf? 2. In how many different ways can 4 different presents can be given to 4 children? 3. In how many ways can the letters of the word CLOSE be arranged? 4. In how many ways can the letters of the word POCKET be arranged? [24] 5. In how many ways can the letters of the word SECTION be arranged? 6. In how many ways can the letters of the word GRADIENT be arranged? [720] [5040] 7. How many four-digit numbers can be formed using the digits 2,3,7and 9 without repetition? [40320] 8. How many five-digit numbers can be formed using the digits 2,3,7, 8 and 9 without repetition? [24] 9. How many different passwords can be formed using the letters A, B, C and digits 8 and 9 without repetition? 10.How many different passwords can be formed using the letters E, F, G, H and symbols α, β and λ without repetition? [144] Permutations & Combinations 2
3 6.1.3 Permutations of n different objects taken r at a time. 1. In how many ways can 6 students be seated on 3 chairs arranged in a row? 2. There are 8 contestants in a drawing contest. How many possible ways for the first three places to be won by the contestants? 3. How many ordered arrangements are there if we want to arrange 5 out of 12 basket players? [336] 4. An examination paper has six questions. The students need only to answer four of them. In how many different ways can the students answer the 4 questions? [95040] 5. In how many different ways can 3 of the letters in the word PROGRAM be arranged without repetition? 6. In how many different ways can 4 of the letters in the word COMPUTER be arranged without repetition? [360] 7. In how many different ways can 2 of the letters in the word VISTA be arranged without repetition? [210] [1680] 8. In how many different ways can half of the letters in the word MODULE be arranged without repetition? 9. How many four-digit numbers can you form using the digits 2, 3, 5, 6 and 7? [20] 10.How many five-digit numbers can you form using the digits 2, 3, 4, 6, 7, 8 and 9? [2520] Permutations & Combinations 3
4 6.1.4 Permutations of n different objects under given conditions. 1. In how many ways can all the letters in the word the first letter is vowel? 2. In how many ways can all the letters in the word the first letter is consonant? [2160] 3. In how many ways can all the letters in the word the first and last letters are vowels? [2880] 4. In how many ways can all the letters in the word the three letters in the middle are vowels? [720] 5. In how many ways can all the letters in the word the first letter is a vowel and last letter is a consonant? [144] 6. In how many ways can all the letters in the word the first letter is E and last letter is T? [1440] 7. In how many ways can all the letters in the word the vowels placed adjacent to each other? 8. In how many ways can all the letters in the word the vowels separated from each other? [720] [4320] Permutations & Combinations 4
5 9. How many five-digit even numbers that can be formed using the digits 2, 3, 4, 6 and 7 without repetition. 10. How many five-digit odd numbers that can be formed using the digits 2, 3, 4, 6 and 7 without repetition. 11. How many five-digit numbers that are greater than can be formed using the digits 2, 3, 4, 6 and 7 without repetition. [72] [48] 12. How many five-digit numbers that are smaller than can be formed using the digits 2, 3, 4, 6 and 7 without repetition. 13. In how many ways can 2 boys and 3 girls be arranged in a line if the two boys must be at the front and the back of the line respectively? [72] 14. In how many ways can 2 boys and 3 girls be arranged in a line if the two boys must be at the first two position of the line respectively? [48] different cards are to be arranged in a row. In how many ways can all the cards be arranged if 2 particular cards must be next to each other? different cards are to be arranged in a row. In how many ways can all the cards be arranged if 3 particular cards must be next to each other? [10080] [4320] Permutations & Combinations 5
6 6.1.5 Permutations of n different objects taken r at a time under given conditions. 1. How many four-letter codes can be formed using the such that the first letter is vowel? 2. How many four-letter codes can be formed using the such that the first letter is consonant? [240] 3. How many four-letter codes can be formed using the such that the first letter and the last letter are vowels? [600] 4. How many four-letter codes can be formed using the such that the first letter and the last letter are consonants? [40] 5. How many four-letter codes can be formed using the such that the first letter is a vowel and the last letter is a constant? [400] 6. How many four-letter codes can be formed using the such that the first letter is S and the last letter is D? [200] 7. How many five-letter codes can be formed using the such that the vowels placed adjacent to each other? [20] 8. How many four-letter codes can be formed using the such that the vowels separated from each other? [720] [4320] Permutations & Combinations 6
7 9. How many four-digit even numbers that can be formed using the digits 2, 3, 4, 6 and 7 without repetition. 10. How many four-digit odd numbers that can be formed using the digits 2, 3, 4, 6 and 7 without repetition. 11. How many four-digit numbers that are greater than can be 3000 formed using the digits 2, 3, 4, 6 and 7 without repetition. [72] 12. How many four-digit numbers that are smaller than can be 6000 formed using the digits 2, 3, 4, 6 and 7 without repetition. [48] [96] boys and 4 girls are to be seated in a row of 5 chairs. Find the number of ways they can be seated if the first and the last are boys. [72] boys and 4 girls are to be seated in a row of 5 chairs. Find the number of ways they can be seated if the first and the last are girls. [48] 15. Calculate the number of ways to arrange 3 different Mathematics books and 4 different Sciences books on a bookrack if the Mathematics books have to be placed together. [288] 16. Calculate the number of ways to arrange 3 different Mathematics books and 4 different Sciences books on a bookrack if the Sciences books have to be placed together. [720] [576] Permutations & Combinations 7
8 6.2.1 Combinations of r different objects chosen from n different objects. 1. In how many ways can 3 prefects be chosen from a group of 8 students? 2. In how many ways can 4 teachers be chosen from a group of 10 teachers for an expedition? [56] 3. A committee of 5 members are to be chosen from 12 candidates. In how many ways can the committee be formed? 4. A team of 5 players are to be chosen from 12 players. In how many ways can the team be formed? [210] [792] 5. Lim is given a chance to choose 4 subjects from the 8 subjects offered in a course. Find the number of combinations she can make? [792] 6. Wahid is given a chance to choose 3 courses from the 8 courses offered in a college. Find the number of combinations he can make? 7. A school has a committee of 3 male teachers and 2 female teachers. Find the number of different such committees that can be formed from 8 male teachers and 11 female teachers? [70] 8. The Parents-Teachers Association of a school has a committee of 3 teachers and 5 parents. Find the number of different such committees that can be formed from 10 teachers and 15 parents? [56] 9. [3080] 10. A B C D A B C D [360360] The diagram shows 4 letters and 3 digits. A code is made up of 3 letters and 2 digits. Find the number of ways it can be. The diagram shows 4 letters and 3 digits. A code is made up of 2 letters and 1 digits. Find the number of ways it can be. [18] Permutations & Combinations 8
9 6.2.2 Combinations of r different objects chosen from n different objects under given conditions. 1. They are a couple in a group of 10 people. In how many ways can a team of 5 people be formed from the group if the couple must be included? 2. They are monitor and assistant monitor in a group of 20 students. In how many ways can a committee of 8 people be formed from the group if the monitor and assistant monitor must be included? 3. Find the number of ways of choosing 4 letters including the letter O from the word OBJECT. [56] 4. Find the number of ways of choosing 4 letters including the letter A from the word IMAGE. [18564] [10] 5. Find the number of ways of choosing at least 3 students from 5 students to represent their class in a Mathematics Quiz. 6. Find the number of ways of choosing at least 1 colour from 3 colours to paint the poster. [4] [6554] 7. In a class of 8 boys and 6 girls, 4 boys and 3 girls or 3 boys and 4 girls are to be chosen to form a committee. In how many ways can the committee be formed? 8. In a group of 10 men and 8 women, 5 men and 4 women or 4 men and 5 women are to be chosen to form a committee. In how many ways can the committee be formed? [7] [2240] students are divided into three groups that consists of 5, 3 and 2 students respectively. Find the number of ways of forming the three groups. [29400] students are divided into three groups that consists of 5, 4 and 3 students respectively. Find the number of ways of forming the three groups. [8820] [27720] Permutations & Combinations 9
10 11.A committee that consists of 6 members is to be selected from 7 teachers and 5 parents. Find the members of different committees that can be formed if the number of teachers must same as the number of parents. 12.A committee that consists of 8 members is to be selected from 9 men and 7 women. Find the members of different committees that can be formed if the number of men must same as the number of women. 13. A committee that consists of 6 members is to be selected from 7 teachers and 5 parents. Find the members of different committees that can be formed if it consists of not more than 2 teachers. [350] [4410] 14.A committee that consists of 8 members is to be selected from 9 men and 7 women. Find the members of different committees that can be formed if it consists of not more than 2 men. [112] 15. A committee that consists of 6 members is to be selected from 7 teachers and 5 parents. Find the members of different committees that can be formed if it consists of at least 4 teachers. [261] 16.A committee that consists of 8 members is to be selected from 9 men and 7 women. Find the members of different committees that can be formed if it consists of at least 6 men. [462] 17. A committee that consists of 6 members is to be selected from 7 teachers and 5 parents. Find the members of different committees that can be formed if it consists of at most 2 teachers. [2025] 18.A committee that consists of 8 members is to be selected from 9 men and 7 women. Find the members of different committees that can be formed if it consists of at most 2 men. [112] [261] Permutations & Combinations 10
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