Simple Counting Problems

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1 Appendix F Counting Principles F1 Appendix F Counting Principles What You Should Learn 1 Count the number of ways an event can occur. 2 Determine the number of ways two or three events can occur using the Fundamental Counting Principle. 3 Determine the number of ways n elements can be arranged. 4 Determine the number of ways n elements can be taken r at a time. Why You Should Learn It Counting principles can be useful when solving real-life application problems. For instance, in Exercise 52 on page F10, you can use counting principles to determine the total number of softball league games played. 1 Count the number of ways an event can occur. Simple Counting Problems This appendix contains a brief introduction to some of the basic counting principles. You will see that much of probability has to do with counting the number of ways an event can occur. Examples 1, 2, and 3 describe some simple cases. Example 1 A Random Number Generator A random number generator (on a computer) selects an integer from 1 to 30. Find the number of ways each event can occur. a. An even integer is selected. b. A number that is less than 12 is selected. c. A prime number is selected. d. A perfect square is selected. a. Because half of the numbers from 1 to 30 are even, this event can occur in 15 different ways. b. The positive integers that are less than 12 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Because this set has 11 members, you can conclude that there are 11 different ways this event can occur. c. The prime numbers between 1 and 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Because this set has 10 members, you can conclude that there are 10 different ways this event can occur. d. The perfect square numbers between 1 and 30 are 1, 4, 9, 16, 25. Because this set has five members, you can conclude that there are five different ways this event can occur.

2 F2 Appendix F Counting Principles Example 2 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. A piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box. Then, a second piece of paper is drawn from the box, and its number is written down. Finally, the two numbers are added together. How many different ways can a total of 12 be obtained? To solve this problem, count the different ways that a total of 12 can be obtained using two numbers between 1 and 8. First number Second number 12 After considering the various possibilities, you can see that this equation can be satisfied in the following five ways. First Number: Second Number: So, a total of 12 can be obtained in five different ways. Solving counting problems can be tricky. Often, seemingly minor changes in the statement of a problem can affect the answer. For instance, compare the counting problem in the next example with the problem in Example 2. Example 3 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two pieces of paper are drawn from the box, and the numbers on them are written down and totaled. How many different ways can a total of 12 be obtained? To solve this problem, count the different ways that a total of 12 can be obtained using two different numbers between 1 and 8. First number Second number 12 After considering the various possibilities, you can see that this equation can be satisfied in the following four ways. First Number: 4578 Second Number: 8754 So, a total of 12 can be obtained in four different ways. Examples 2 and 3 differ in that the random selection in Example 2 occurs with replacement, whereas the random selection in Example 3 occurs without replacement, which eliminates the possibility of choosing two 6 s. When doing such exercises, be sure to note if selection is with or without replacement.

3 Appendix F Counting Principles F3 2 Determine the number of ways two or three events can occur using the Fundamental Counting Principle. Counting Principles The first three examples in this section are considered simple counting problems in which you can list each possible way that an event can occur. When it is possible, this is always the best way to solve a counting problem. However, some events can occur in so many different ways that it is not feasible to write out the entire list. In such cases, you must rely on formulas and counting principles. The most important of these is called the Fundamental Counting Principle. Fundamental Counting Principle Let E 1 and E 2 be two events. The first event E 1 can occur in m 1 different ways. After E 1 has occurred, E 2 can occur in m 2 different ways. The number of ways that the two events can occur is m 1 m 2. The Fundamental Counting Principle can be extended to three or more events. For instance, the number of ways that three events E 1, E 2, and E 3 can occur is m 1 m 2 m 3. Example 4 Applying the Fundamental Counting Principle How many two-letter words can be made from the English alphabet? The English alphabet contains 26 letters. So, the number of possible two-letter words is Example 5 Applying the Fundamental Counting Principle Telephone numbers in the United States have 10 digits. The first three are the area code and the next seven are the local telephone number. How many different telephone numbers are possible within each area code? (A telephone number cannot have 0 or 1 as its first or second digit.) There are only eight choices for the first and second digits because neither can be 0 or 1. For each of the other digits, there are 10 choices. Area code ( ) - So, by the Fundamental Counting Principle, the number of local telephone numbers that are possible within each area code is ,400,000. Local number

4 F4 Appendix F Counting Principles 3 Determine the number of ways n elements can be arranged. Permutations One important application of the Fundamental Counting Principle is in determining the number of ways that n elements can be arranged (in order). An ordering of n elements is called a permutation of the elements. Definition of Permutation A permutation of n different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on. Example 6 Listing Permutations How many permutations are possible for the letters A, B, and C? The possible permutations of the letters A, B, and C are as follows. A, B, C B, A, C C, A, B A, C, B B, C, A C, B, A So, six permutations are possible. Example 7 Finding the Number of Permutations of n Elements How many permutations are possible for the letters A, B, C, D, E, and F? First position: Any of the six letters Second position: Any of the remaining five letters Third position: Any of the remaining four letters Fourth position: Any of the remaining three letters Fifth position: Any of the remaining two letters Sixth position: The one remaining letter So, the numbers of choices for the six positions are as follows. Permutations of six letters By the Fundamental Counting Principle, the total number of permutations of the six letters is ! 720. The formula for the number of permutations of n elements is given on the next page.

5 Appendix F Counting Principles F5 Number of Permutations of n Elements The number of permutations of n elements is given by n n n!. So, there are n! different ways that n elements can be ordered. Example 8 Finding the Number of Permutations How many ways can you form a four-digit number using each of the digits 1, 3, 5, and 7 exactly once? One way to solve this problem is simply to list the number of ways. 1357, 1375, 1537, 1573, 1735, , 3175, 3517, 3571, 3715, , 5173, 5317, 5371, 5713, , 7153, 7315, 7351, 7513, 7531 Another way to solve the problem is to use the formula for the number of permutations of four elements. By that formula, there are 4! 24 permutations. Example 9 Finding the Number of Permutations How many ways can you form a six-digit number using each of the digits 1, 2, 3, 4, 5, and 6 exactly once? By the formula for the number of permutations of six elements, there are 6! 720 permutations. Example 10 Finding the Number of Permutations You are a supervisor for 11 different employees. One of your responsibilities is to perform an annual evaluation for each employee, and then rank the performances of the 11 different employees. How many different rankings are possible? Because there are 11 different employees, you have 11 choices for the first ranking. After choosing the first ranking, you can choose any of the remaining 10 for the second ranking, and so on. Rankings of 11 employees So, the number of different rankings is 11! 39,916,800.

6 F6 Appendix F Counting Principles 4 Determine the number of ways n elements can be taken r at a time. Combinations When counting the number of possible permutations of a set of elements, order is important. The final topic in this section is a method of selecting subsets of a larger set in which order is not important. Such subsets are called combinations of n elements taken r at a time. For instance, the combinations A, B, C and B, A, C are equivalent because both sets contain the same three elements, and the order in which the elements are listed is not important. So, you would count only one of the two sets. A common example of how a combination occurs is a card game in which the player is free to reorder the cards after they have been dealt. Do you remember the following riddle? How many different ways can two of the three (wolf, goat, and cabbage) be left on the shore? A farmer wants to take a wolf, a goat, and a cabbage across a river. Unattended, the wolf will eat the goat, and the goat will eat the cabbage. The farmer can take only one of the three on each trip across the river. How can the farmer take all three safely across the river? There are three possible combinations that can be left on the shore. wolf, goat, wolf, cabbage, goat, cabbage Of the three combinations, the farmer can only leave the wolf alone with the cabbage when he crosses the river. Example 11 Combinations of n Elements Taken r at a Time In how many different ways can three letters be chosen from the letters A, B, C, D, and E? (The order of the three letters is not important.) The following subsets represent the different combinations of three letters that can be chosen from the five letters. A, B, C A, B, D A,B,E A,C,D A,C,E A, D, E B, C, D B, C, E B, D,E C,D, E From this list, you can conclude that there are 10 different ways that three letters can be chosen from five letters. The order is not important. So, a set such as B, C, A is not chosen, because it is represented by the set A, B, C. The formula for the number of combinations of n elements taken r at a time is as follows. Number of Combinations of n Elements Taken r at a Time The number of combinations of n elements taken r at a time is n C r n! n r!r!.

7 Appendix F Counting Principles F7 Study Tip When solving problems involving counting principles, you need to be able to distinguish among the various counting principles in order to determine which is necessary to solve the problem correctly. To do this, ask yourself the following questions, and use the indicated principle if the answer is yes. 1. Is the order of the elements important? Permutation 2. Are the chosen elements a subset of a larger set in which order is not important? Combination 3. Does the problem involve two or more separate events? Fundamental Counting Principle Note that the formula for n C r is the same one given for binomial coefficients. To see how this formula is used, consider the counting problem given in Example 11. In that problem, you need to find the number of combinations of five elements taken three at a time. So, n 5, r 3, and the number of combinations is 5 C 3 which is the same as the answer obtained in Example 11. Example 12 Combinations of n Elements Taken r at a Time The Boston Market restaurant chain offers individual rotisserie chicken meals with two side items. Customers can select side items from a list of 12. How many different rotisserie chicken meals are available if a customer wishes to order two different side items? (Source: Boston Market, Inc.) Use the formula for the number of combinations of 12 elements taken two at a time. 12 C 2 5! 2!3! 10 12! 10!2! So, there are 66 different meals available. Example 13 Combinations of n Elements Taken r at a Time A standard poker hand consists of five cards dealt from a deck of 52. How many different poker hands are possible? (After the cards are dealt, the player may reorder them, and therefore order is not important.) Use the formula for the number of combinations of 52 elements taken five at a time, as follows. 52 C 5 52! 47!5! ,598,960 So, there are almost 2.6 million different hands.

8 F8 Appendix F Counting Principles Appendix F Exercises Review Concepts, Skills, and Problem Solving Keep mathematically in shape by doing these exercises before the problems of this section. Properties and Definitions 1. Which of the following functions is (are) exponential? Explain. f x 5x 2 gx 25 x In Exercises 7 10, solve the equation. (Round the result to two decimal places.) 7. 3 x e x log 2 x lnx Explain why e 2x2 e 2 e x 2. Logarithms and Exponents In Exercises 3 6, rewrite the equation in exponential form. 3. log 4. log ln ln Problem Solving 11. Depreciation After t years, the value of a car that cost $22,000 is given by Vt 22, t. Sketch a graph of the function and determine when the value of the car is $15, Decay Carbon 14 has a half-life of 5730 years. You start with 10 grams of this isotope. How much remains after 3000 years? Solving Problems Random Selection In Exercises 1 6, find the number of ways the specified event can occur when one or more marbles are selected from a bowl containing 10 marbles numbered 0 through 9. See Examples One marble is drawn and its number is even. 2. One marble is drawn and its number is prime. 3. Two marbles are drawn one after the other. The first is replaced before the second is drawn. The sum of the numbers is Two marbles are drawn one after the other. The first is replaced before the second is drawn. The sum of the numbers is Two marbles are drawn without replacement. The sum of the numbers is Two marbles are drawn without replacement. The sum of the numbers is 7. Random Selection In Exercises 7 16, find the number of ways the specified event can occur when one or more marbles are selected from a bowl containing 20 marbles numbered 1 through One marble is drawn and its number is odd. 8. One marble is drawn and its number is even. 9. One marble is drawn and its number is prime. 10. One marble is drawn and its number is greater than One marble is drawn and its number is divisible by One marble is drawn and its number is divisible by Two marbles are drawn one after the other. The first is replaced before the second is drawn. The sum of the numbers is Two marbles are drawn one after the other. The first is replaced before the second is drawn. The sum of the numbers is 15.

9 Appendix F Counting Principles F9 15. Two marbles are drawn without replacement. The sum of the numbers is Three marbles are drawn one after the other. Each marble is replaced before the next is drawn. The sum of the numbers is Staffing Choices A small grocery store needs to open another checkout line. Three people who can run the cash register are available, and two people are available to bag groceries. In how many different ways can the additional checkout line be staffed? See Examples 4 and Computer System You are in the process of purchasing a new computer system. You must choose one of three monitors, one of two computers, and one of two keyboards. How many different configurations of the system are available to you? 19. Identification Numbers In a statistical study, each participant was given an identification label consisting of a letter of the alphabet followed by a single digit (0 is a digit). How many distinct identification labels can be made in this way? 20. Identification Numbers How many identification labels (see Exercise 19) can be made using one letter of the alphabet followed by a two-digit number? 21. License Plates How many distinct automobile license plates can be formed by using a four-digit number followed by two letters? 22. License Plates How many distinct license plates can be formed by using three letters followed by a three-digit number? 23. Three-Digit Numbers How many three-digit numbers can be formed in each of the following situations? (a) The hundreds digit cannot be 0. (b) No repetition of digits is allowed. (c) The number cannot be greater than Three-Letter Codes How many three-letter codes can be formed in each of the following situations? (a) The middle letter must be a vowel. (b) No repetition of letters is allowed. (c) The code is a palindrome. (A palindrome is a string of letters that reads the same front-to-back as it does back-to-front.) 25. Toboggan Ride Five people line up on a toboggan at the top of a hill. Only two of the five are willing to sit in the front seat. In how many ways can they be seated? 26. Task Assignment Four people are assigned to four different tasks. One of the four people is not qualified for the first task. In how many ways can the assignments be made? 27. Taking a Trip Five people are taking a trip in a car. Two sit in the front seat and three in the back seat. Three of the people agree to share the driving. In how many different ways can the five people sit? 28. Aircraft Boarding Eight people are boarding an aircraft. Three have tickets for first class and board before those in economy class. In how many different ways can the eight people board the aircraft? 29. Permutations List all the permutations of the letters X, Y, and Z. See Examples 6 and Permutations List all the permutations of the letters A, B, C, and D. 31. Permutations List all the permutations of two letters selected from the letters A, B, and C. 32. Permutations List all the permutations of two letters selected from the letters A, B, C, and D. 33. Seating Arrangement In how many ways can five children be seated in a single row of five chairs? See Examples Seating Arrangement In how many ways can six people be seated in a six-passenger car? 35. Posing for a Photograph In how many ways can four children line up in one row to have their picture taken? 36. Batting Order In how many ways can nine baseball players be put into a batting lineup? 37. Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 40, inclusive) is selected. How many different lock combinations are possible? 38. Access Code An access code will unlock a door when the right choice of five numbers (from 0 to 9, inclusive) is selected. How many different access codes are possible? 39. Work Assignments Eight workers are assigned to eight different tasks. In how many ways can this be done assuming there are no restrictions in making the assignments?

10 F10 Appendix F Counting Principles 40. Work Assignments Out of eight workers, five are selected and assigned to different tasks. In how many ways can this be done if there are no restrictions in making the assignments? 41. Choosing Officers From a pool of 10 candidates, the offices of president, vice-president, secretary, and treasurer will be filled. Each of the 10 candidates can hold any office. In how many ways can the offices be filled? See Examples Time Management Study There are eight steps in accomplishing a certain task, and these steps can be performed in any order. Management wants to test each possible order to determine which is the least time-consuming. (a) How many different orders will have to be tested? (b) How many different orders will have to be tested if one step in accomplishing the task must be done first? (The other seven steps can be performed in any order.) 43. Number of Subsets List all the subsets with two elements that can be formed from the set of letters A, B, C, D, E, F. 44. Number of Subsets List all the subsets with three elements that can be formed from the set of characters A, B, C, 1, 2, Committee Selection Three students are selected from a class of 20 to form a fundraising committee. In how many ways can the committee be formed? 46. Committee Selection In how many ways can a committee of six be formed from a group of 25 people? 47. Menu Selection A group of four people go out to dinner at a restaurant. There are nine entrees on the menu, and the four people decide that no two will order the same entree. In how many ways can the four people order from the nine entrees? 48. Menu Selection A group of six people go out to dinner at a restaurant. There are 12 entrees on the menu, and the six people decide that no two will order the same entree. In how many ways can the six people order from the 12 entrees? 49. Test Questions A student may answer any nine questions from the 12 questions on an exam. Determine the number of ways the student can select the nine questions. 50. Test Questions A student may answer any three questions from the 10 questions on an exam. Determine the number of ways the student can select the three questions. 51. Basketball Lineup A high school basketball team has 15 players. Use a graphing calculator to determine the number of ways the coach can choose five players in the starting lineup. (Assume that each player can play each position.) 52. Softball League Six churches form a softball league. Each team must play every other team twice during the season. What is the total number of league games played? 53. Job Applicants An employer interviews six people for four openings in the company. Four of the six people are women, and all six are qualified. How many ways could the employer fill the four positions if (a) the selection is random? (b) exactly two women are selected? 54. Defective Units A shipment of 10 microwave ovens contains two defective units. In how many ways can a vending company purchase three of these units and receive (a) all good units? (b) two good units? (c) one good unit? 55. Group Selection Four people are to be selected from four couples. In how many ways can this be done if (a) there are no restrictions? (b) one person from each couple is selected? 56. Player Selection Three three-player teams compete in a basketball tournament. After the tournament, a three-player all-star team is to be selected. In how many ways can this be done if (a) there are no restrictions? (b) one player from each team is selected?

11 Appendix F Counting Principles F Geometry Eight points are located in the coordinate plane such that no three lie on the same line. How many different triangles can be formed having their vertices as three of the eight points? 61. Octagon 62. Decagon 58. Geometry Three points that are not on the same line determine three lines. (see figure). How many lines are determined by seven points, no three of which are on a line? Relationships As the size of a group of people increases, the number of possible two-person relationships increases dramatically (see figure). In Exercises 63 and 64, determine the number of two-person relationships in a group of each given size. 63. (a) 3 (b) 6 (c) (a) 4 (b) 8 (c) 12 Geometry In Exercises 59 62, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices of a polygon is called a diagonal of the polygon.) 59. Pentagon 60. Hexagon n = 2 n = 3 n = 4 n = 5 Figure for 63 and 64 Explaining Concepts 65. State the Fundamental Counting Principle. 66. When you use the Fundamental Counting Principle, what are you counting? 67. Give examples of a permutation and a combination. 68. Without calculating the numbers, determine which is greater: the number of combinations of 10 elements taken six at a time or the number of permutations of 10 elements taken six at a time. Explain. 69. Without calculating the numbers, determine which is greater: the number of combinations of 10 elements taken seven at a time or the number of combinations of 10 elements taken three at a time. Explain.

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