11.7 The Fundamental Counting Principle and Permutations 731 41. Club Officers If a club consists of 10 members, how many different arrangements of president, vice president, and secretary are possible? 720 42. Winning the Trifecta The trifecta at most racetracks consists of selecting the first-, second-, and third-place finishers in a particular race in their proper order. If there are seven entries in the trifecta race, how many tickets must you purchase to guarantee a win? 210 43. Flag Messages Five different colored flags will be placed on a pole, one beneath another. The arrangement of the colors indicates the message. How many messages are possible if five flags are to be selected from nine different colored flags? 15,120 44. Jay Leno’s Garage Jay Leno owns 117 motorcycles of which he must select 4 to line up in his garage for his television show, Jay Leno’s Garage. How many different arrangements of 4 motorcycles are possible? 177,928,920 53. In how many ways can the digits in the number 4,568,865 be arranged? 630 54. In how many ways can the digits in the number 2,142,332 be arranged? 420 Challenge Problems/Group Activities 55. Ranked Choice Voting In an election for mayor, voters are asked to rank three of the 5 candidates, indicating their first choice, second choice, and third choice. What is the minimum number of ballots that must be cast to guarantee that at least two ballots are the same? 61 ballots 56. Ranked Choice Voting On a ballot, each committee member is asked to rank three of eight candidates for recommendation for promotion, giving first, second, and third choices (no ties). What is the minimum number of ballots that must be cast to guarantee that at least two ballots are the same? 337 ballots 57. Scrabble Nancy is playing Scrabble and has seven different letters. She decides to test each five-letter permutation. If each permutation takes 5 sec, how long will it take Nancy to check all the permutations? 12,600 sec, or 3.5 hr 58. Scrabble Cathy is playing Scrabble and has seven letters of which three are identical and two are identical. Cathy decides to test each seven-letter permutation. If each permutation takes 5 sec, how long will it take Cathy to check all the permutations? 2100 sec, or 35 min Recreational Mathematics 59. Stations There are eight bus stations from town A to town B. How many different single tickets must be printed so that a passenger may purchase a ticket from any station to any other station? 56 A B 60. Bus Loop How many tickets with different points of origin and destination can be sold on a bus line that travels a loop with 25 stops? 600 Jumble In Exercises 61 and 62, read the Recreational Math box on page 728 and determine a) the number of different possible arrangements of the letters given. b) the word when the letters are placed in the proper order. 61. HEICOC a) 360 b) CHOICE 62. DUTENST a) 2520 b) STUDENT Research Activity 63. ISBN When a book is published, it is assigned a 10-digit code number called the International Standard Book Number (ISBN). Do research and write a report on how this coding system works. m Jay Leno Ian West/PA Images/Alamy Stock Photo 45. Skiing Event A skiing event has 16 participants for the slalom. The 5 participants with the fastest speeds will be listed, in the order of their speed, on the leader board. How many different ways are there for the five names to be listed? 524,160 46. Swimming Event A swimming event has fifteen participants. The 4 swimmers with the fastest speeds will be listed, in order, on the leader board. How many different ways are there for the names to be listed? 32,760 47. Determine the number of permutations of the letters of the word EDUCATION. 362,880 48. Determine the number of permutations of the letters in the word IMPORTANCE. 3,628,800 49. In how many ways can the letters in the word STATISTICS be arranged? 50,400 50. In how many ways can the letters in the word MANAGEMENT be arranged? 226,800 51. In how many ways can the letters in the word KISSIMMEE be arranged? 22,680 52. In how many ways can the letters in the word CHATTAHOOCHEE be arranged? 32,432,400
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