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12.1 Permutations When a question says ‘how many arrangment’..... Think BOXES •For each space we have a box •In the box write down how many options can go into it •Multiply these numbers e.g 1 e.g 1 (i) How many arrangements can be made of the letters of the word FROG taking two letters at a time Long way 4 3 = 12 FR RF OF GF FO RO OR GR =3 FG RG OG GO (i) How many start with a vowel 1 o 3 Deal with the restriction first e,g, 2 The digits 0,1,2,3,4,5 are to form a three digit code. A code cannot begin with a 0 and no digit can be repeated. How many codes can be formed ? 5 No o 5 4 = 100 Deal with the restriction first e.g 3 (i) How many different numbers can be formed from the digits 2, 3, 4,5, 6, if each of the digits can be used only once in each number? (ii) How many of the numbers are less than 400? (iii) How many of the numbers are divisible by 5? (iv) How many of the numbers are less than 400 and divisible by 5? e.g 3 (i) (ii) (i) How many different numbers can be formed from the digits 2, 3, 4, 5, 6, if each of the digits can be used only once in each number? (ii) How many of the numbers are less than 400? (iii) How many of the numbers are divisible by 5? (iv) How many of the numbers are less than 400 and divisible by 5? 5 2 4 4 2,3 (iii) 4 3 3 2 2 1 1 = 120 = 48 Deal with the restriction first 3 2 1 1 = 24 5 (iv) 2 2,3 3 2 1 1 5 = 12 e.g 4 A code consists of a four-digit number which is formed from the digits 3 to 9 inclusive. No digit can occur more than once in the code. (i) Write down the smallest possible four-digit code. (ii) How many different codes are possible? (iii) How many of the four-digit codes are greater than 6000? (iv) How many of the four-digit codes are divisible by 2?c e.g 4 A code consists of a four-digit number which is formed from the digits 3 to 9 inclusive. No digit can occur more than once in the code. (i) Write down the smallest possible four-digit code. (ii) How many different codes are possible? (iii) How many of the four-digit codes are greater than 6000? (iv) How many of the four-digit codes are divisible by 2?c (i) 3 4 5 6 (ii) 7 6 5 4 = 840 (iii) 4 6 5 4 = 480 6,7,8,9 (iv) 6 Deal with the restriction first 5 4 3 4,6,8 = 360 e.g 5 Three boys and two girls are seated in a row as a group. In how many different ways can the group be seated if (i) there are no restrictions on the order of seating (ii) there must be a boy at the beginning of the row (iii) there must be a boy at the beginning of the row and a boy at the end of the row (iv) the two girls must be seated beside each other? e.g 5 Three boys and two girls are seated in a row as a group. In how many different ways can the group be seated if (i) there are no restrictions on the order of seating (ii) there must be a boy at the beginning of the row (iii) there must be a boy at the beginning of the row and a boy at the end of the row (iv) the two girls must be seated beside each other? (i) (ii) 5 3 Boy (iii) 4 3 3 2 2 1 1 = 120 = 72 Deal with the restriction first 3 3 2 1 4 3 2 1 Boy (iv) 4 2 Boy = 120 = 24 = 24 x 2 = 48 to be seated together take one of the seats away Then multiply the answer by 2 as the girls could swap places with each other