Permutations (Part A)

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1 Permutations (Part A) A permutation problem involves counting the number of ways to select some objects out of a group. 1

2 There are THREE requirements for a permutation. 2

3 Permutation Requirements 1. The n objects are all different/distinguishable. 2. No object can be repeated. 3. Order makes a difference 3

4 With permutations, every little detail matters. 4

5 Permutations are for arrangements where the order of the objects matters. Alice, Bob and Charlie is different from Charlie, Bob and Alice. 5

6 Telephone numbers are a good example of a permutation. For example, the phone number is different from

7 A permutation is just the fundamental counting principle expressed as a formula. 7

8 Permutation Formula The number of permutations of n different objects taken r at a time is given by:, n r "n permutation of r" 8

9 Note: When r = n, all of the objects in a group are selected and arranged in a specific order. 9

10 Example Anna, Marie and Brian line up at a banking machine. How many different arrangements are there? List all the possible ways these three people can stand in line. 10

11 A list of possible arrangements are: There are 6 possible arrangements. A Anna M Marie B Brian AMB MAB BAM ABM MBA BMA 11

12 Sample Problems 1. Evaluate. a) 5 P 3 b) 10 P 4 c) 8 P 8 12

13 2. a) Solve for n. np 2 = 30 13

14 2. b) Solve for r. 5P r = 20 14

15 3. Suppose 1 st, 2 nd and 3 rd place prizes are to be awarded to a group of 8 trumpeters. How many ways are there to award 3 prizes to 8 people? 15

16 4. How many different 5 digit numbers can be made using the numbers 1, 2, 3, 4, 5, 6, 7 by only using each number once? 16

17 5. How many ways can a president, vice president, a secretary and treasurer be selected from a class of 25 students? 17

18 6. How many ways can 7 books be arranged on a shelf if they are selected from 10 different books? 18

19 7. From 25 raffle tickets, 5 tickets are to be selected in order. The first ticket wins $250, the second $200, the third $150, the fourth $100 and the fifth $50. How many ways can these prizes be awarded. 19

20 So far, all examples of permutations have been distinguishable permutations. This means that all the objects are different from one another. 20

21 Permutations: Part B Suppose you had the letters a, b, c, and d and were asked to form all words using these four letters. How many words can be formed? 21

22 Your answer would be different if the letters you had to work with were a, a, b, c. This would be an example of a permutation that is non distinguishable. 22

23 In the 24 words formed, half will appear the same because we cannot distinguish between the two a's. We would only have 12 unique words. 23

24 Permutation with Repetition The number of permutations of n objects in which n 1 are alike, n 2 alike, etc., is: where 24

25 Sample Problems 1. How many different words can be formed using the letters of the word LIBBY? 25

26 2. What is the total number of permutations of the letters in the word BANANA? 26

27 3. Find the number of different ways of placing 16 balls in a row given that 4 are black, 3 are green, 7 are red, and 2 are blue. 27

28 4. Miss Sherrard's hockey team has 20 players consisting of 12 forwards, 6 defence, and 2 goalies. How many ways can you arrange the players? 28

29 5. How many different words can be formed using the letters of the word MISSISSIPPI? 29

30 Specific Positions Frequently when arranging items, a particular position must be occupied by a particular item. The easiest way to approach these questions is by analyzing how many possible ways each space can be filled. 30

31 Sample Problems 1. How many ways can Adam, Beth, Charlie and Doug be seated in a row if Charlie must be in the second chair? 31

32 2. How many ways can you order the letters of KITCHEN if the arrangements must start with a consonant and end with a vowel? 32

33 3. How many ways can you order the letters of UMBRELLA if the arrangements must begin with exactly two L's? 33

34 4. How many ways can you order the letters of TORONTO if it begins with exactly two O's? NOTE: Exactly two O's means the first 2 letters must be O, and the third must NOT be an O. Don't forget repetitions. 34

35 Items Always Together Sometimes, certain items must be kept together. To do these questions, you must treat the joined items as if they were only one object. 35

36 Sample Problems 1. How many arrangements of the word ACTIVE are there if C and E must always be together? 36

37 2. How many ways can 3 math books, 5 chemistry books and 7 physics books be arranged on a shelf if the books of each subject must be kept together? 37

Sec. 4.2: Introducing Permutations and Factorial notation

Sec. 4.2: Introducing Permutations and Factorial notation Permutations: The # of ways distinguishable objects can be arranged, where the order of the objects is important! **An arrangement of objects in