permutation and combination

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Your Teacher :

Your Teacher Pn Saripah Ahmad Sek. Men Sains Muzaffar Syah Melaka Email : sahmozac@gmail.com Assalamulaikun

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PERMUTATIONS AND COMBINATIONS  MULTIPLICATION PRINCIPLE/RULE  PERMUTATIONS  COMBINATIONS

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MULTIPLICATION PRINCIPLE/RULE

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If an operation can be carried out r ways and another operation can be carried out in s ways , then the number of ways to carry out both operations is r x s .

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EXAMPLE 1 There are 3 different roads to travel from P to town Q and 4 different roads to travel from town Q to town R. Calculate the number of ways a person can travel from town P to town R via town Q. ANSWER 3 x 4 = 12

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EXAMPLE 2 Team A which consists of 6 players wishes to have pig-pong matches with team B which consists of 7 players. Calculate the number of different single matches that can be held. ANSWER 6 x 7 = 42

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PERMUTATIONS

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FACTORIAL 3! = 3 X 2 X 1 n! = n x (n-1) x (n- 2)X…3 x 2 x 1 6! = 6 x 5 x 4 x 3 x 2 x 1 TIPS: 0! = 1

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 The number of permutations of n different objects is n! .  The number of permutations of n different objects taken r at a time is  The number of permutations of n different objects taken all at a time is FORMULAE TO BE USED The order of the objects in the chosen set is taken into consideration.

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EXAMPLE 1 Calculate the number of ways to arrange 9 books of different titles on a bookrack. ANSWER 9 8 7 6 5 4 3 2 1

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EXAMPLE 2 Calculate the number of 5-digit numbers that can be formed using the digits 1, 2, 3, 4 and 5, without repetition. ANSWER 5 4 3 2 1

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EXAMPLE 3 For the word ‘MINUTES’, calculate the number of different arrangements that can be formed if the arrangements have to begin with a vowel. ANSWER 5 4 3 2 1 I/U/E 6 3 No of Arrangements or

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EXAMPLE 4 Calculate the number of 5-digit odd numbers that can be formed by arranging the digits 3, 5, 6, 8 and 9, without repetitions. ANSWER 4 2 1 3/5/9 3 No of Arrangements or 3

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EXAMPLE 5 Calculate the number of 4-digit numbers that exceed 7000 can be formed by arranging the digits 5, 6, 7 and 8, without repetitions. ANSWER 2 1 7/8 2 No of Arrangements 3

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EXAMPLE 6 Calculate the number of 3-digit numbers that can be formed using the digits 2, 4, 6, 8 and 9, without repetitions. ANSWER 3 5 No of Arrangements 4

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EXAMPLE 7 Using the digits 3,4,5,6 and 7, without repetitions, calculate: (a) the number of 4-digit numbers that are greater than 5000 that can be formed. (b) the number of 4-digit odd numbers that are less than 5000 that can be formed. ANSWER 1 (a) 3 3 4 2 (b) 2 1 3 2 3 3/5/7 4 3 5/7 2 3 + 5/6/7

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EXAMPLE 8 Calculate the number of ways to arrange 3 different cooking books and 4 different health books on a bookrack if the 3 cooking books have to be placed together. ANSWER C C C H H H C C C H H H H H C C C H H H H C C C H H H H C H H H H C C

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EXAMPLE 9 7 boys and 3 girls are to be seated in a row. Calculate the number of ways they can be seated if the 3 girls want to be seated together. ANSWER B B B G G G B B B B

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EXAMPLE 10 A group of 4 men and 3 ladies are to be seated in a row for a photographing session. If the men and ladies want to be seated alternately, calculate the number of different arrangements. ANSWER B G B G B G B 4 3 2 1 3 2 1 4! X 3!

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COMBINATIONS

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FORMULA TO BE USED  The number of combinations of r objects chosen from n objects is The order of the objects in the chosen set is not taken into consideration.

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EXAMPLE 1 Calculate the number of ways to select a pair of double players from a group of 7 badminton players. ANSWER

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EXAMPLE 2 A committee that consists of 6 teachers is to be chosen from 7 male teachers and 5 female teachers. Find the number of different committees can be formed if (a) there is no restriction, (b) 3 male teachers and 3 female teachers are required in the committee. ANSWER (a) (b)

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EXAMPLE 3 Jack wants to choose 7 compact discs from 5 local compact discs and 9 foreign compact discs. Calculate the number of different choices if his choices must have at least 3 local compact discs. ANSWER LOCAL, 5 FOREIGN, 9 3 4 4 3 5 2 7

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EXAMPLE 2 A tennis team that consists of 8 students is to be chosen from a group of 7 Form Four students and 6 Form Five students. Calculate the number of teams that can be formed if (a) the team must consist of exactly 5 Form Four students. (b) the number of Form Four students must be more than the number of Form Five students. ANSWER (a) (b) Form 4, 7 Form 5, 6 7 1 6 2 5 3 8 + + =

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DO ALL THE EXERCISES IN THE MODULE

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