Counting and Probability

0838 ch0_p639-693 0//007 0:3 PM Page 633 CHAPTER 0 Counting and Probability The design below is like a seed puff of a dandelion just before it is dispersed by the wind. The design shows the outcomes from rolling two dice, one after the other. The number of outcomes can be counted in order to assign probability. From the odds in gambling and buying lottery tickets to the chances of success in trying a new treatment for a disease, probability is fundamental to our culture. In this chapter, we look at ways of counting outcomes from various activities, ways of choosing sets of objects, and the probabilities of the outcomes and sets of objects. 6, 6 6, 6 6, 4 6, 3 4 6, 3 6,,, 3, 4,,, 6 3 4 6 6,,, 3 3 4, 4, 6, 6 6, 6, 4 3, 4, 3,, 4 6 3 4 4, 6 4, 4, 4, 4 4, 3 4, 3 6 3, 6 3 4 3, 3 3, 4 3, 3, 3, FIGURE 633

0838 ch0_p639-693 0//007 0:4 PM Page 634 634 CHAPTER 0 COUNTING AND PROBABILITY 0. Counting Outcomes Objectives Identify a random experiment, an outcome, a sample space, and an event. Construct a one-, two-, and three-stage tree diagram. Apply the multiplication principle of counting to find the number of outcomes. Apply the addition principle of counting to find the number of outcomes. WARM-UP Evaluate these expressions.. 0 0 0 6 6 6. 6 6 6 0 0 0 3. 9 6 6 6 0 0 0 4. (6 7). (6 7) 6. (6 7) A FRIEND ONCE NEEDED to choose between two universities with similar financial requirements but situated in different parts of the country. A career counselor advised her to list the pros and cons for each and, if she was still undecided, flip a coin, assigning one school to heads and one school to tails. After the flip, she was to analyze her feelings about the outcome and base her decision on her feelings. What do you think about this decision-making process? Basic Definitions FIGURE Six faces of a die An experiment is an activity with an observable outcome or result. In a random experiment, the outcome is not known in advance. A sample space is the set of all possible outcomes. An event is a subset of a sample space. An event may include one or more outcomes. In the university decision-making process, flipping the coin is a random experiment. Heads and tails are the sample space. In Example, we roll a die (one of a pair of dice). Figure shows the six faces. EXAMPLE Applying definitions For the random experiment of rolling a single six-sided die, tell which of the following is an event containing two outcomes, the sample space, and an event containing one outcome. a. The set {,, 3, 4,, 6} b. Obtaining a on the top face of the die c. Obtaining either a or a 6 on the top face of the die a. The sample space b. An event containing one outcome c. An event containing two outcomes We will assume that all experiments in the next four sections are random. To make the mathematics more apparent, the experiments involve dice, coins, birth order, spinners, and familiar daily experiences. EXAMPLE Exploring a sample space List all the possible outcomes (the sample space) for rolling two dice, one red and one green.

0838 ch0_p639-693 0//007 0:3 PM Page 63 SECTION 0. COUNTING OUTCOMES 63 The red cube has six faces (,, 3, 4,, 6). The green cube has six faces (,, 3, 4,, 6). Figure 3 shows the 36 outcomes in the sample space. FIGURE 3 THINK ABOUT IT : How can we be confident that all outcomes are shown in Figure 3? Tree Diagrams A tree diagram is a systematic way to show the sample space. The branches in the tree diagram in Figure 4a show the two outcomes from flipping a coin. The outcomes, or sides of a coin, are labeled H (head) and T (tail). Assume that no coin lands on its edge. When we flip the coin a second time, we draw two more branches on the end of each prior branch, as shown in Figure 4b. H H T H T H T (a) FIGURE 4 We read the tree by tracing out each branch, in this case from left to right. There are four outcomes from flipping a coin twice: HH, HT, TH, TT. THINK ABOUT IT : How many branches would be needed for a tree diagram of the roll of one die? How many branches would be needed in total if the tree diagram also included the roll of a second die? Figure 4a is a one-stage tree; Figure 4b is a two-stage tree. In Example 3, we build a three-stage tree. EXAMPLE 3 Building a tree diagram Draw a tree diagram showing the possible outcomes (H for head, T for tail) from flipping a coin 3 times. Use the tree to list the outcomes. (b) T In Figure, we start with two branches for the first flip. We then add a set of two branches to each prior branch for the second flip, and do the same for the third flip.

0838 ch0_p639-693 0//007 0:3 PM Page 636 636 CHAPTER 0 COUNTING AND PROBABILITY H H T To list the outcomes, we record the letters along each branch of the tree. There are eight outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. THINK ABOUT IT 3: Group the outcomes in Example 3 by number of tails. H T H T The tree diagram in Figure has two outcomes (H and T) at each branching. We now look at branches for the spinners in Figure 6. T H H T T H T FIGURE FIGURE 6 EXAMPLE 4 Building a tree diagram a. Draw a tree diagram for the outcomes from turning first spinner (with the numbers) and then spinner (with the letters). b. Use the tree diagram to list the outcomes in the sample space. c. How many stages are there in the tree diagram? a. The tree diagram in Figure 7 is read from top to bottom. Assume that any pointer landing on a line is spun again. 3 A B C D A B C D A B C D FIGURE 7 b. There are twelve outcomes: A, B, C, D, A, B, C, D, 3A, 3B, 3C, 3D. c. The tree has two stages. THINK ABOUT IT 4: Would the size of the sample space change if we used spinner first and spinner second? How are the outcomes the same? Different? For experiments with large numbers of outcomes, such as rolling dice with six faces, listing the sample space or drawing a tree diagram can be tedious. In many cases, we need only the number of outcomes. Two ways we can find the number of outcomes from an activity are with the multiplication principle of counting and the addition principle of counting.

0838 ch0_p639-693 0//007 0:3 PM Page 637 SECTION 0. COUNTING OUTCOMES 637 Multiplication Principle of Counting MULTIPLICATION PRINCIPLE OF COUNTING If a first event A can happen in a ways and, after it or separately, a second event B can happen in b ways, then the number of ways in which both A and B can happen is a b. The multiplication principle of counting can be extended to three or more events. THINK ABOUT IT : Which phrase in the statement of the multiplication principle of counting says that order or arrangement is important? To apply the multiplication principle of counting, we look for distinct objects (different colored dice) or for one event following another event (a coin flipped repeatedly). We build a product by including an underline for each distinct object or event. EXAMPLE Counting outcomes A standard die has six faces. Use the multiplication principle of counting to find the number of outcomes for each of the following. a. Rolling two dice, one red and one green. Compare the results with those in Example. b. Rolling three dice, one red, one green, and one white. The dice are distinct. We draw an underline for each die and use the multiplication principle. a. 6 6 36; there are six outcomes for red and six outcomes for green. This number agrees with the result in Example. b. 6 6 6 6 3 6; there are six outcomes each for red, green, and white. THINK ABOUT IT 6: What information in Example tells us that the dice are distinct? EXAMPLE 6 Counting outcomes How many outcomes are there for flipping a coin a. once? b. twice? c. 3 times? d. 4 times? a. Two outcomes: head or tail We start with an underline for each flip and use the multiplication principle. b. 4 outcomes A factor of for each flip c. 3 8 outcomes A factor of for each flip d. 4 6 outcomes A factor of for each flip The multiplication principle of counting is used to count the number of outcomes in several practical applications. EXAMPLE 7 Counting outcomes In 963, the post office introduced zip codes. Twenty years later, in 983, the post office added a four-digit extension. All ten numbers (0 to 9) are used in zip codes, and repetition is permitted that is, numbers may be repeated. a. How many different zip codes are possible with a -digit zip code? b. How many different zip codes are possible with a 9-digit zip code? We use the multiplication principle. a. 0 0 0 0 0 0 00,000 There are 0 digits for each position in the code. b. 0 0 0 0 0 0 0 0 0 0 9 There are 0 digits for each position in the code.

0838 ch0_p639-693 0//007 0:3 PM Page 638 638 CHAPTER 0 COUNTING AND PROBABILITY EXAMPLE 8 Counting outcomes California automobile license plates used to have 3 numbers followed by 3 letters. They were changed to have a digit ( or ) followed by 3 letters (A to Z) followed by 3 digits (0 to 9). a. How many different plates were possible with 3 numbers followed by 3 letters? b. How many different plates are possible with a lead digit of or followed by 3 letters and then 3 numbers? c. How many plates would be possible if the license could start with any digit from to 9? There are 0 digits and 6 letters available to fill the different positions on the license plate. We use the multiplication principle. a. 0 0 0 6 6 6 0 3 6 3 7,76,000 b. 6 6 6 0 0 0 6 3 0 3 3,,000 c. 9 6 6 6 0 0 0 9 6 3 0 3 8,84,000 In Examples 7 and 8, we could repeat numbers and letters in the zip codes and license plates. In Example 9, the objects are used up, so no repetition is permitted. EXAMPLE 9 Counting outcomes While on vacation, you buy 6 different postcards to send to 6 friends. How many ways are there to send the postcards? You have a choice of 6 cards to send to the first friend, cards to send to the next friend, 4 cards to send to the next friend, and so forth. By the multiplication principle, 6 4 3 70 There are 70 ways to send 6 postcards to 6 friends. Addition Principle of Counting FIRST ADDITION PRINCIPLE OF COUNTING Let A and B be two events whose sample spaces contain no common outcomes. Then if event A can happen in a ways and event B can happen in b ways, the number of ways in which either A or B can happen is a b. The addition principle of counting can be extended to three or more events. To apply the addition principle, we look for phrases meaning there is no outcome both in the sample space for event A and in the sample space for event B. Events are mutually exclusive if their sample spaces contain no common outcome. We will consider events that are not mutually exclusive in Section 0.4. EXAMPLE 0 Counting outcomes When you roll two dice, how many ways is it possible to roll a. a sum of 6? b. a sum of 6 or a sum of 7? c. a sum of 6 and a sum of 7? The sample space for rolling two dice was shown in Example. a. The dice in Figure 8 show a sum of 6. Each outcome happens one way, so by the addition principle there are ways to obtain a sum of 6. FIGURE 8

0838 ch0_p639-693 0//007 0:3 PM Page 639 SECTION 0. COUNTING OUTCOMES 639 b. From part a, there are ways to have a sum of 6. Figure 9 shows 6 ways to have a sum of 7. There are no outcomes in common. By the addition principle, there are 6 ways to have a sum of 6 or a sum of 7. FIGURE 9 c. Zero; there are no pairs of dice whose top faces add to both 6 and 7 at the same time. EXAMPLE Counting outcomes A coin is flipped three times. a. How many outcomes have exactly heads or exactly 3 heads? b. How many outcomes have exactly head and exactly tail? a. From Example 3, there are three outcomes with exactly heads: HHT, HTH, THH. There is one outcome with exactly 3 heads: HHH. (The outcomes are mutually exclusive.) By the addition principle, there are 3 4 ways to have exactly heads or exactly 3 heads. b. Zero; the third flip must result in either heads or tails, so there are no outcomes with exactly head and exactly tail. Making a Distinction between the Multiplication Principle and the Addition Principle Use the multiplication principle when order is implied that is, when one event happens and then another event. When one event or another event (but not both) is a possible outcome, use the addition principle. EXAMPLE Counting outcomes Pizza Palace offers 6 kinds of vegetable toppings (onions, tomatoes, etc.) and 7 kinds of meat toppings (sausage, salami, etc.). In ordering a pizza, how many different ways could you select a. a vegetable topping or a meat topping? b. a vegetable topping and a meat topping? c. three different vegetable toppings? a. 6 7 3 choices Addition principle b. 6 7 4 choices Multiplication principle c. 6 4 0 choices Multiplication principle EXAMPLE 3 Combining the addition and multiplication principles Pizza Palace has flavors of soda and kinds of pizza crusts to go with the toppings in Example. In placing an order, how many ways could you select a. a soda and a vegetable or a meat topping? b. a crust and a vegetable or a meat topping? c. a soda and a crust and a vegetable or a meat topping? a. (6 7) 6 Add the topping choices and then multiply by soda choices. b. (6 7) 6 Add the topping choices and then multiply by crust choices. c. (6 7) 30 Add the topping choices and then multiply by the soda and crust choices. THINK ABOUT IT 7: Is part a of Example 3 equivalent to a soda and a vegetable topping or to a soda and a meat topping?

0838 ch0_p639-693 0//007 0:3 PM Page 640 640 CHAPTER 0 COUNTING AND PROBABILITY ANSWER BOX Warm-up:. 7,76,000. 3,,000 3. 8,84,000 4. 6. 6 6. 30 Think about it : Each number on the red die is matched with each number on the green die. Think about it : Six branches; six branches on each of the six branches, for 36 in total; see Figure, p. 633. Think about it 3: No tails: HHH; one tail: HHT, HTH, THH; two tails: HTT, THT, TTH; three tails: TTT. Think about it 4: No; both orders have outcomes. The numbers and letters in the outcomes are reversed. Think about it : The phrase after it or separately. Think about it 6: The colors: red, green, white. Think about it 7: Yes; (6 7) 6 7. Which property of real numbers does this illustrate? 0. Exercises. List the sample space for birth orders (outcomes) in terms of girls (G) or boys (B) for families with a. children b. 3 children c. 4 children. List the sample space in terms of heads (H) or tails (T) for flipping a coin 4 times. Organize your list to group outcomes together by the number of heads. 3. List the sample space for possible outcomes (first, second, third) of a golf tournament playoff among Sergio Garcia, Vijay Singh, and Tiger Woods. 4. List the sample space for possible outcomes (first, second, third) of a tennis tournament playoff among Lindsay Davenport, Serena Williams, and Venus Williams.. Draw a tree diagram showing the outcomes from flipping a coin and then rolling a die. 6. Draw a tree diagram showing the outcomes from rolling a die and then flipping a coin. 7. Draw a tree diagram for the results from turning a spinner with three equal regions (A, B, C) and then flipping a coin. 8. Draw a tree diagram for flipping a coin and then turning a spinner with four equal regions (A, B, C, D). 9. How many birth orders by gender (boy or girl) are there for the following numbers of children? a. children b. 6 children c. 0 children d. n children 0. How many outcomes are there from flipping a coin the following numbers of times? a. b. 6 c. d. n. What is the same about the birth orders by gender of n children and the outcomes from n coin flips?. How are the sample spaces for Exercises and different from the sample spaces for Exercises 3 and 4? 3. How many different 9-digit Social Security numbers are possible? 4. The first 3 digits of a Social Security number represent the state (or territory) where the application was made. How many different 3-digit numbers are possible?. A test has 0 multiple-choice questions with choices a, b, c, and d. How many different answer keys are possible? 6. A test has true/false questions. How many different answer keys are possible? 7. A round combination padlock has the numbers 0 to 39 (see the figure). How many 3-number combinations are possible a. if numbers can be repeated? b. if numbers cannot be repeated? Blue numbers are core exercises.

0838 ch0_p639-693 0//007 0:3 PM Page 64 SECTION 0. COUNTING OUTCOMES 64 8. A house security number key pad has the digits 0 to 9. How many 4-number combinations are possible a. if numbers can be repeated? b. if numbers cannot be repeated? 9. Emma has packed skirts and 3 pairs of slacks. In dressing from her suitcase, how many ways can she choose to wear either a skirt or a pair of slacks? 0. Professor Anita has chosen 4 fiction and 6 nonfiction books for her Latin Studies course. In assigning books to read for the course, how many ways can she choose the first book?. Use the results in part c of Exercise to find the number of birth orders for 4 children that have a. exactly 3 boys or exactly boys. b. exactly 3 girls and exactly boys. c. exactly girl or exactly boy. d. exactly 3 girls and exactly boy.. Use the results in Exercise to find the number of outcomes from flipping a coin 4 times that have a. exactly heads and exactly tails. b. exactly heads or exactly 3 tails. c. exactly head or exactly tail. d. exactly head and exactly tail. 3. How many different radio or television station call names can be made if the first letter must be either a K or a W and the name must contain a. 3 letters (as in KEX or WOR)? b. 4 letters (as in KLCC or WGBH)? c. How many 3- or 4-letter station names can be made altogether? 4. Suppose a roadside stand has 3 types of cones, type of dish, 33 flavors of ice cream, and 8 flavors of sherbet. How many different outcomes are there for a. a dish with a scoop of ice cream or sherbet? b. a -scoop cone? c. a -scoop cone? d. a dish with a scoop of ice cream and a scoop of sherbet? e. a 3-scoop cone with different flavors of sherbet?. An animal rescue house has 7 cats and dogs. How many ways can Jay pick a. a cat and a dog? b. a cat or a dog? c. a cat for himself and a cat for his son? d. a dog for himself and a dog for his daughter? 6. The Breakfast Diner has eggs (6 ways), meat (4 choices), toast (4 choices), and potatoes ( ways). How many ways can Kay order a. eggs or toast? b. eggs, meat, toast, and potatoes? c. eggs, meat, and toast? d. eggs, meat, and potatoes? e. eggs and meat with a choice of toast or potato? f. eggs with a choice of meat or toast or potato? 7. A college offers sociology, psychology, 3 economics, and history courses that count toward the Social Sciences lower-division requirement. Courses cannot be repeated. In completing requirements, how many ways can a student take a. psychology or history course? b. sociology and economics course? c. 3 history and sociology course? d. history or sociology course? e. 3 sociology courses? f. psychology and history or economics course? 8. Explain how the distributive property permits us to rewrite the questions in parts e and f of Exercise 6 and part f of Exercise 7 and obtain the same solutions. 9. The figure below shows the sample space for rolling two dice. Because some of the events in this exercise are not mutually exclusive, use this sample space to count the number of ways to obtain each of the following outcomes. Blue numbers are core exercises.

0838 ch0_p639-693 0//007 0:3 PM Page 64 64 CHAPTER 0 COUNTING AND PROBABILITY a. The same number on both dice b. One or two s c. A and a 3 d. A or a 3 e. One 3 or two 3s 30. A standard deck of cards has 4 suits (spades, hearts, diamonds, and clubs) of 3 cards each. The cards are numbered to 0, jack, queen, king. How many ways are there to a. give card each to players? b. give diamond each to players? c. draw a diamond and then a card of a different suit? d. draw a 9? e. draw a (ace) or a face card ( jack, queen, king)? 3. How might the sample space of 36 outcomes for rolling two dice (Exercise 9) be used to list the sample space of 6 outcomes for rolling three dice (as found in Example )? 3. How many possible 4-digit extensions to the zip code are there? Why is this number multiplied by, not added to, the number of -digit zip codes to find the total number of possible 9-digit codes? 33. Telephone Numbers a. Until cell phones became popular, telephone area codes were limited to a set of 3 numbers with a 0 or in the center position, such as 0 or 66. No area codes start with a 0 or. How many 3-digit codes were possible? b. The center position now can be any number. How many 3-digit area codes are possible? c. The 3-digit area code is followed by a 7-digit local telephone number. If neither the area code nor the local number can start with a 0 or a, how many telephone numbers are possible? d. In all area codes, the telephone number - is reserved by the phone company for directory assistance. Other phone numbers are used in television shows and movies. How many 0-digit phone numbers are available if all numbers are excluded? (Think carefully.) e. Write and solve a problem of your own having to do with -800 and related phone numbers. Projects 34. Dice Sums and Differences Use the sample space for the rolling of two dice from Exercise 9. Organize your answers to a and b in one table. Organize your answers to c and d in another table. a. List the sums of the top faces on the pairs of dice. b. How many times does each sum appear? c. What are all the possible differences between the top faces on the two dice? Give the absolute values of the differences. d. How many times does each difference appear? e. Are all sums equally likely (do they each have the same chance of happening)? What sum is most likely? Least likely? f. Are all differences equally likely? What difference is most likely? Least likely? g. How are the outcomes (sums and differences) different? h. How are the outcomes alike? 3. Playing Cards and Not Mutually Exclusive Events Use the standard deck of cards described in Exercise 30. a. List the outcomes in the event heart. b. List the outcomes in the event. c. List the outcomes in the event of hearts. d. What makes the outcomes in or heart not mutually exclusive? Do Section 0.3 before completing parts e to i. e. What is the probability of a? f. What is the probability of a heart? g. What is the probability of a or a heart? h. Why isn t the probability in part g equal to 7, the sum of the probabilities in parts e and f? i. Suggest a formula for probability when the events are not mutually exclusive. Try your formula on the probability of a face card ( jack, queen, king) or a club.

0838 ch0_p639-693 0//007 0:4 PM Page 643 SECTION 0. PERMUTATIONS AND COMBINATIONS 643 0. Permutations and Combinations Objectives Define and evaluate factorials. Find permutations of a set of objects. Find the number of permutations of n objects taken r at a time without repetition. Find combinations of a set of objects. Find the number of combinations of n objects taken r at a time without repetition. Distinguish among settings involving the multiplication principle of counting, permutations, and combinations. WARM-UP Evaluate these expressions.. 0 9 8 7. 3 3. 4 3 4. 4 3. What shortcut might you use to multiply 6 4 3? IN THIS SECTION, we look at formal ways of counting sets of objects. Factorials The multiplication principle, introduced in Section 0., suggests how to find the number of ways a deck of playing cards can be arranged: 0 49 48 47 46 4 44 43 4. Well, you get the idea; the last three numbers are 3. The entire product of the numbers is called a factorial. FACTORIAL If n is a positive integer, then n factorial is defined as n! n(n )(n )(n 3) 3 where! and 0!. Because of their connection to the natural number e, factorials were introduced in Section 8.6, on page 6. The repetition here is minimal. EXAMPLE Evaluating factorials Find the values of these factorials. a. 3! 3 b. 4! 4 3 c. 7! a. 6 b. 4 c. 7 6 4 3 040 EXAMPLE Comparing factorial expressions Are the expressions on each side of the equals sign equivalent? Explain why or why not. a.! 4! b. 7! 7 6! c.! 0 49! d. 3! 3 e.! f. 6! 6!!

0838 ch0_p639-693 0//007 0:3 PM Page 644 644 CHAPTER 0 COUNTING AND PROBABILITY a. Equivalent: b. Equivalent: 4! 4 3! 7 6! 7 6 4 3 7! c. Not equivalent; the right side is missing as a factor. d. Not equivalent; the right side is missing a factorial sign on the. e. Equivalent:!! f. Equivalent: 4 4 3 3 6 4 3 4 3 6 6 THINK ABOUT IT : If a and b are integers, is a!b! a rational number? To evaluate the factorial n!, enter the number n and choose factorial with MATH PRB 4 :! ENTER. Repeat Example on a calculator. When factorials appear in the numerator and denominator, we look for simplifications. EXAMPLE 3 Evaluating factorials Simplify these expressions. a. 4! 0! b. 4! c.!! d. 0! 4! 48! e. 4!! f. 0!! g. (4 )! h. (0 4)! i. 0! 4! j. 0! 4! a. 4 3 4 0! by definition b. 4 4! by definition c.! 0! 0! 0!! d. 4! 48! 6 0 49 48! 4 3 48! 0 49 4 3 0 4 9 4 3 3 7 49 70,7 Note the 48!48!. Simplify if you choose. Multiply with a calculator. e. 4!! 4 6 Factorials and multiplication precede addition in the order of operations. f. 0!! By definition g. (4 )! 6! 70 This is not the same as part e.

0838 ch0_p639-693 0//007 0:3 PM Page 64 SECTION 0. PERMUTATIONS AND COMBINATIONS 64 h. (0 4)! 6! 70 i. 0! 4! 3,68,800 4 3,68,776 This is not the same as part h. j. 0 4!! 8,800,00 4 Follow the order of operations. When you see the directions simplify these expressions, you should now include changing factorial expressions into products of factors, eliminating common factorials and factors from numerators and denominators, and calculating the resulting product. Factorials appear in multiplication principle of counting solutions when choices cannot be repeated. EXAMPLE 4 Applying factorials A college newspaper has 4 students interested in the positions of editor, news editor, advertising editor, and production editor. No one can hold more than one position. How many ways are there to fill the 4 positions? State the answer as a factorial. The newspaper has a choice of 4 students for editor, then 3 for news editor, for advertising editor, and student for production editor. Using the multiplication principle, we have 4 3 4 There are 4! ways to fill the four positions. THINK ABOUT IT : Which sentence or phrase in Example 4 indicates that choices cannot be repeated? Permutations and Combinations In Section 0., we counted outcomes using sample spaces, tree diagrams, the multiplication principle, and the addition principle. Two related ways of classifying and counting objects are in terms of permutations and combinations. Factorials are used in calculating the numbers of both permutations and combinations. Several of the example and exercise settings in this section are from Section 0.. PERMUTATIONS In Example 4, the number of people wanting positions on the newspaper was equal to the number of positions. In Example, there are more applicants than positions; thus, we might expect more choices for each position than in Example 4. EXAMPLE Exploring outcomes The college newspaper has 0 students interested in positions. How many ways are there to fill the 4 positions of editor, news editor, advertising editor, and production editor? Each person can hold no more than one position. The 4 positions to be filled are denoted by 4 underlines. The newspaper has a choice of 0 students for editor, 9 for news editor, 8 for advertising editor, and 7 for production editor. Using the multiplication principle, we have 0 9 8 7 040 There are 040 ways to fill the 4 positions.

0838 ch0_p639-693 0//007 0:3 PM Page 646 646 CHAPTER 0 COUNTING AND PROBABILITY Example gives the number of permutations. PERMUTATION A permutation* is an ordered arrangement of all or part of a set of n objects, where the arrangement contains r objects without repetition. EXAMPLE 6 Recognizing a permutation Describe how the situation in Example fits the definition of permutation state r and n and tell how ordered and without repetition were satisfied. In Example, we chose r 4 students from a set of n 0 students. The different jobs provided order. The fact that each person could hold no more than one position satisfied without repetition. EXAMPLE 7 Recognizing a permutation Which of the following are permutation (and multiplication principle) problems? Which apply the multiplication principle only? Solve each problem. List the outcomes for the permutations. a. How many ways are there to choose and set aside number from the set {,, 3, 4}? b. How many ways are there to make a -digit number using the set {,, 3, 4} if digits may be repeated? c. How many ways are there to make a -digit number using the set {,, 3, 4} if digits may not be repeated? d. How many ways are there to select 3 numbers from the set {,, 3, 4} if we replace the numbers each time? a. Permutation because setting aside guarantees no repetition; 4 ways:,, 3, 4. b. Multiplication principle because of repeated numbers; 4 4 6 ways. c. Permutation because a -digit number implies order and there is no repetition; 4 3 ways:, 3, 4,, 3, 4, 3, 3, 34, 4, 4, 43. d. Multiplication principle because the number is replaced each time; 4 4 4 64 ways. PERMUTATION FORMULA The number of permutations of n objects taken r at a time (r n) is written np r and calculated with the formula n! np r (n r)! EXAMPLE 8 Using permutation notation Write the permutation problems in Examples 4,, and 7 in n P r notation and apply the formula to find the number of permutations. Observe how a!a! helps simplify the expressions. In Example 4, 4! 4P 4 4! 4 3 4 (4 4)! 0! In Example, 0! 0P 4 (0 4)! 0 9 6 7 6! 8 040! 0 9 8 7 6 4 3 6 4 3 or *In this introduction, we will consider only permutations without repetition. Other variations, such as circular permutations, exist.

0838 ch0_p639-693 0//007 0:3 PM Page 647 SECTION 0. PERMUTATIONS AND COMBINATIONS 647 In part a of Example 7, 4! 4P 4 3 or 4 3! 4 (4 )! 3 3! In part c of Example 7, 4! 4P 4 3 (4 )! or 4 3!! EXAMPLE 9 Evaluating n P r notation Find the value of each expression. a. 0P 3 b. P 9 c. P 4 d. 6P 6 e. 6P 0 Observe how a!a! helps simplify the expressions. 0! a. 0P 3 (0 3)!! b. P 9 ( 9)! 0 9 8 7! 0 7 9 8 70! 0 9 8 7 6 4 79,833,600! c. P 4 ( 4)! 0 49 48! 0 49 6,497,400 6! d. 6P 6 6! 6 4 3 70 (6 6)! 0! 6! e. 6P 0 6! (6 0)! 6! 0 9 8 7 6 4 3! 3! 48! To evaluate permutation notation, enter the number n, press MATH, select PRB : npr, enter r, and press ENTER. Redo Example 9 on a calculator, first using factorials and then using the permutation symbol n P r. The results for part a are shown in Figure 0. FIGURE 0 EXAMPLE 0 Applying permutations Explain why the problem describes a permutation and then answer the question: How many 3-letter words can be formed from the letters A, B, C, D, and E if no letter may be repeated? The problem describes a permutation because we are selecting r 3 letters from a set of n letters to be arranged in left to right order without repetition. The number of words is! P 3 4 ( 3)!! 3! 4 3 60

0838 ch0_p639-693 0//007 0:3 PM Page 648 648 CHAPTER 0 COUNTING AND PROBABILITY EXAMPLE Counting outcomes with the multiplication principle Repeat Example 0 using the multiplication principle. We write 3 underscores to represent the letters in each word. There is no repetition of letters, so we have a choice of letters for the first position, 4 for the second, and 3 for the third. 4 3 60 THINK ABOUT IT 3: How does a comparison of Examples 0 and show why the permutation formula is reasonable? COMBINATIONS Next we explore how removing the order requirement changes the number of outcomes when we select objects from a set. EXAMPLE Recognizing order Which of the following problem settings involve permutations? Give the number of outcomes and list the outcomes for each. In all four parts, assume that there is no repetition. a. The number of ways to choose a president and vice president from {a, b, c} b. The number of different -person committees from {a, b, c} c. The number of different sets of 3 books on a table from {a, b, c} d. The number of ways to stand 3 books on a shelf from {a, b, c} a. Permutation because the offices imply order; 6 ways: ab, bc, ca, ac, cb, ba b. Not a permutation because a committee has no order (committee ab is the same as committee ba); 3 committees: ab, ac, bc c. Not a permutation because a set of 3 books on a table has no order; set: {a, b, c} d. Permutation because there is a left to right order; 6 ways: abc, acb, bac, bca, cab, cba The settings in Example in which order does not matter are combinations. COMBINATION A combination is a subset of r objects selected without regard to order from a set of n objects. COMBINATION FORMULA The number of combinations of n objects taken r at a time (r n) is written nc r and calculated with the formula n! nc r (n r)!r! EXAMPLE 3 Using combination notation Write parts b and c of Example in combination notation and use the formula to find the number of combinations. In part b of Example, 3! 3! 3 3C (3 )!!!!! 3! In part c of Example, 3! 3! 3! 3C 3 (3 3)!3! 0! 3! 3!

0838 ch0_p639-693 0//007 0:3 PM Page 649 SECTION 0. PERMUTATIONS AND COMBINATIONS 649 EXAMPLE 4 Exploring arrangements Is the number of different -person selections from the set {a, b, c} both a combination and a permutation? Compare 3 C with 3 P. Yes; when only one person is being selected, there is no other arrangement of that one person. The condition about order is meaningless in this case. 3! 3C 3! 3 (3 )!!! 3! 3P 3! 3 (3 )!! 3C has the same value as 3 P. One-person selections appear to be both a combination and a permutation. EXAMPLE Evaluating n C r notation Find the value of each expression. a. 4C 4 b. 4C 3 c. 4C d. 4C e. 4C 0 f. 0C 0 g. 0C 4! 4! 4! a. 4C 4 (4 4)!4! 0! 4! 4! 4! 4 b. 4C 3 3! 4 3! 4 (4 3)!3!! 3! 3! 4! c. 4C 4 3! 6 (4 )!!! 4! d. 4C 4 3! 4 (4 )!! 3! 4! 4! e. 4C 0 (4 0)!0! 4! 0! 0! f. 0C 0 (0 0)!0! 0! 0! 0! g. 0C 0 9! 0 (0 )!! 9! 9! To evaluate combination notation on a calculator, enter n, press MATH, select PRB 3 : ncr, enter r, and press ENTER. Redo Example on a calculator first using factorials and then using the combination symbol n C r. The results for part a are shown in Figure. FIGURE THINK ABOUT IT 4: In Example, which combination formulas have the same value? Why? EXAMPLE 6 Applying combinations The Toy Store has 00 different toys. How many different selections of 3 toys are possible?

0838 ch0_p639-693 0//007 0:3 PM Page 60 60 CHAPTER 0 COUNTING AND PROBABILITY The selection does not imply order. And buying or 3 of the same toy probably would not make sense. 00! 00C 3 (00 3)!3! 0 0 0 00 99 9 3 00 000 999 998! 998! 3 00 00 333 66,666,00 There are 66,666,00 different selections possible. Redo Example 6 on a calculator. The results are shown in Figure. FIGURE EXAMPLE 7 Applying combinations A 7-member hiring committee is to be selected from 4 administrators, faculty members, and 4 counselors. a. How many different committees are possible? b. How many different committees are possible if there must be administrators, 4 faculty members, and counselor? a. Nothing is stated about the composition of the committee or the order in which it is selected. We use a combination with n all possible committee members: 3! 3C 7 4,7 (3 7)!7! b. Hint: Choose the administrators first, then the faculty members, then the counselor. 4C 6 ways C 4 36 ways 4C 4 ways Then Choose administrators. Choose 4 faculty members. Choose counselor. 6 36 4 3,760 ways Multiplication principle Special Applications of Combinations: Outcomes from Repeated Trials A trial occurs if we repeat an experiment. Flipping a coin 4 times results in 4 trials, each with the outcome H (head) or T (tail). EXAMPLE 8 Applying combinations a. List the outcomes (H or T) for flipping a coin 4 times. Group the outcomes by number of heads, starting with 4 heads. b. Compare your results with those in parts a to e of Example.

0838 ch0_p639-693 0//007 0:3 PM Page 6 SECTION 0. PERMUTATIONS AND COMBINATIONS 6 a. For 4 flips: HHHH HHHT HHTT HTTT TTTT HHTH HTHT THTT HTHH HTTH TTHT THHH TTHH TTTH THTH THHT b. For 4 flips: The one outcome with 4 heads matches the value for 4 C 4 and for 4 C 0. The total of four outcomes with 3 heads matches the value for 4 C 3 and for 4 C. The total of six outcomes with heads matches the value for 4 C. The total of four outcomes with head matches the value for 4 C and for 4 C 3. The one outcome with 0 heads matches the value for 4 C 0 and for 4 C 4. These pairings suggest that the number of outcomes with r heads in n flips is related to the combination notation n C r. The birth of a child (G or B) and the flip of a coin (H or T) are experiments with two outcomes. The number of outcomes from n trials of such experiments can be summarized in the combination notation n C r. The variable r is the number of heads (or tails) in n flips of a coin or the number of girls (or boys) in n births. This connection between repeated trials and combinations is explored further in Exercise 4 and in Section 0.. THINK ABOUT IT : What can we say about the values of n C a and n C b when a b n? EXAMPLE 9 Applying the n C r formula How many ways are there to obtain heads in flipping a coin 8 times? We have n 8 trials of the flip of a coin and r heads. n! nc r (n r)!r! 8! 8C 8 (8 )!! 6! 7 6! 8 There are 8 ways to obtain heads in flipping a coin 8 times. Redo Example 9 on a calculator. The results are shown in Figure 3. FIGURE 3

0838 ch0_p639-693 0//007 0:3 PM Page 6 6 CHAPTER 0 COUNTING AND PROBABILITY Making a Distinction among the Multiplication Principle, Permutations, and Combinations Recall that in settings where the objects chosen from a set may be repeated (or the drawn object is replaced), the multiplication principle applies, but neither the permutation formula nor the combination formula does. All permutation problems can be solved with the multiplication principle. Table compares the multiplication principle and the permutation formula. TABLE Setting Multiplication Principle Permutation Formula Members of a club Call r with repetition Call r in any order without repetition A collection of books Read r (rereading allowed) Call r without repetition; order counts A group of letters Choose 3 with repetition: Choose 3 without repetition; order counts: {a, b, c, d} {aaa, abc, acb, aac, ddd, bad,...} {abc, acb, bad,...} 4 4 4 64 ways 4 3 4 ways 4! 4P 3 4 (4 3)! A group of numbers Choose with repetition: Choose without repetition; order counts: {,, 3} {,, 3,,, 3, 3, 3, 33} {, 3,, 3, 3, 3} 3! 3 3 9 ways 3P 6 (3 )! TABLE THINK ABOUT IT 6: Are the outcomes that form a permutation in Table a subset of the outcomes counted by the multiplication principle? Table compares permutations and combinations. Setting Permutations Combinations Members of an organization Selecting officers Selecting a committee A collection of books Books lined up on a shelf Books tossed into a backpack A group of letters Three-letter words that can be made: Read r without repetition; order counts {a, b, c, d} {abc, acb, bad, abd, etc.} {abc, abd, acd, bcd} 4! 4! 4P 3 4 4C 3 4 (4 3)! (4 3)!3! EXAMPLE 0 Distinguishing problem types Tell whether each setting involves the multiplication principle, permutations, or combinations. Solve the problem. a. To get an Associate of Arts transfer degree from a two-year college, a student must take 3 quarter courses, plus electives. In a school with perfectly flexible schedules, how many ways could you take 4 of the 3 courses during the first term? Assume no repetition of courses. b. How many ways could the 3 courses be listed in a column on the advisors summary sheet? c. A career planning survey is to be given to the students in 4 courses, drawn at random from the 3 courses. How many ways could the courses be selected?

0838 ch0_p639-693 0//007 0:3 PM Page 63 SECTION 0. PERMUTATIONS AND COMBINATIONS 63 d. Four courses are to be featured (one per page) on pages, 7, 33, and 49 of the class schedule booklet. How many ways could the courses be featured? e. If repeating of courses were permitted, how many ways could 4 of the 3 courses be taken? a. Order is not implied, so this is a combination. (Taking math, writing, economics, and Spanish is the same as taking Spanish, writing, economics, and math.) 3! 3C 4 (3 4)!4! 3,960 b. Order is implied and all courses are listed. 3 3 30 3 3!.633 0 3 Multiplication principle 3P 3.633 0 3 c. Order is not implied, so this is a combination. 3C 4 3,960 d. Order is implied. 3 3 30 9 8! 8! 4 3 3 3 30 9 863,040 Multiplication principle 3P 4 863,040 Permutation e. Repetition is allowed in drawing from a given set of objects. Permutation 3 3 3 3 3 4,048,76 Multiplication principle ANSWER BOX Warm-up:. 040. 6 3. 4 4. 0. 6 times the answer to Exercise 4 Think about it : Yes; the product of integers will be an integer, so both the numerator and the denominator will be integers, satisfying the requirement for a rational number. Think about it : The sentence No one can hold more than one position. Think about it 3: If all n objects were used, the numerator, by the multiplication principle, would be n!. However, only r of the n objects are chosen, so the last (n r) positions of the n! product must be eliminated. This is done in the n P r formula with a division by (n r)!. Think about it 4: 4 C 4 4 C 0 0 C 0, 4 C 3 4 C 4. When r 0 or r n, the denominator in the combination formula will equal the numerator, and the result will be. When two combinations have r values adding to n, the combinations will be equal. Think about it : n C a n C b when a b n. Think about it 6: Yes; each outcome for the permutation is also listed (or implied) in the outcomes for the multiplication principle. 0. Exercises. Evaluate. a. 8! b. 0! c. 0! d. (8 )! e. 8!! f. (8 )!. Evaluate. a. 9! b.! c.! d. (9 )! e. 9!! f. 9!! Blue numbers are core exercises.

0838 ch0_p639-693 0//007 0:3 PM Page 64 64 CHAPTER 0 COUNTING AND PROBABILITY 3. True or false? a. 6! 6 4 b.! 4 3! c.! 0! d. 3 0! is undefined! e. 3!! 6! f. 6! 0 9 8 4. True or false? a. 8! 8 7 6! b. 9! 9 8 7 6 c. 0! 0 d. 3! 3! 3! e.! 4! f. 4!! 8!. Simplify. a.!! b. 3!! 6! c. 0! 4! d. (6 )!! 8! e. f. 0!! (8 )!! 6. Simplify. a. 6!! b. 3!! 6! c.! 0! d. (6 4)!4! 8! e. f.!! (8 3)!3! In Exercises 7 to, tell whether the problem is a factorial and/or a multiplication principle (MP) problem. Answer the question. 7. Brad has tied 6 flies for fly-fishing. How many ways can he use them one at a time? 8. Marie has 4 notebooks. How many ways can she label them for her 4 courses? 9. Sally has 7 pairs of space socks. How many ways can she wear them in a week if she wears one pair each day and may repeat wearings? 0. Zane has 8 books. How many ways can he arrange them on a shelf?. Orville has 7 wing nuts. How many ways can he fasten them on a row of 7 threaded screws?. Julia has 9 favorite menus. How many ways can she use them over a period of 9 dinners if she may repeat the menu? 3. Write in permutation notation: arrangements of 3 letters from the set {a, b, c, d}. 4. Write in permutation notation: arrangements of 3 letters from the set {v, w, x, y, z}.. Write in permutation notation and then list the permutations: arrangements of letters from the set {a, b, c, d}. 6. Write in permutation notation and then list the permutations: arrangements of letters from the set {w, x, y}. In Exercises 7 to 0, evaluate. 7. a. P 4 b. P c. P 8. a. 6P 3 b. 6P c. P 9. a. P 3 b. P 3 c. P 8 0. a. 6P b. P c. P 7. How many ways can 3-letter words be made from the set {L, M, N, O, P} a. if no letter may be repeated? b. if letters may be repeated?. How many ways can books be selected from the set {history, math, chemistry, French, and economics} a. if the first book is returned to the set after reading? b. if the first book is not returned to the set after reading? 3. How many ways can CDs be selected from a set of 8 CDs a. if the first CD is returned to the set after playing? b. if the first CD is not returned to the set after playing? 4. How many ways can 3 substitutions be made from a group of 8 players a. if no substitute can go into the game more than once? b. if a substitute may return to the game?. How many ways can first, second, and third places be selected from show dogs? 6. How many ways can a meeting of 0 people be interrupted by 3 personal cell phone calls? State your assumptions. 7. Write in combination notation and list the outcomes: a combination of 3 letters taken from {w, x, y, z}. 8. Write in combination notation and list the outcomes: a combination of letters taken from {w, x, y, z}. 9. Write in combination notation and list the outcomes: a combination of letters taken from {b, c, d}. 30. Write in combination notation and list the outcomes: a combination of 3 letters taken from {b, c, d, e, f}. Blue numbers are core exercises.

0838 ch0_p639-693 0//007 0:3 PM Page 6 SECTION 0. PERMUTATIONS AND COMBINATIONS 6 In Exercises 3 to 34, evaluate. 3. a. 6C 4 b. 6C c. C 8 3. a. 7C 3 b. 7C c. C 9 33. a. 8C 3 b. C c. C 34. a. 9C b. C c. C In Exercises 3 to 38, tell what word or phrase indicates that a combination is appropriate and then answer the question. 3. How many different sets of 6 numbers can be selected from 48 numbers? 36. How many different purchases of 3 CDs are possible from an artist s collection of? 37. How many different sets of 3 letters can be selected from a set of 0 different letters? 38. How many different 3-person committees can be selected from a department of members? 39. Find the number of combinations, and then select the appropriate explanation from the sentences below. () There are n ways to select object from a set containing n objects. () There is only one way to choose all the elements from the original set. (3) There is only one set with nothing in it, the empty set. a. C 0 b. C c. C 40. In which of the following will the order of the objects matter? Tell whether each is a permutation or a combination setting. a. Choosing officers b. Choosing a discussion group c. Buying books d. Placing books on a shelf In Exercises 4 to 0, tell whether the settings require the multiplication principle (MP), the addition principle (AP), permutations (and the multiplication principle), or combinations. Then evaluate. 4. A college ID has 4 numbers and/or letters drawn randomly from a set of 36 numbers and letters (A to Z, 0 to 9). For example, one ID has ABE6. a. How many different 4-place IDs can be made if repetitions are permitted? b. How many different 4-place IDs can be made if no repetitions are permitted? c. At a school where repetitions are permitted in the IDs, the bookstore decides to give out $0 gift certificates each week for the middle five weeks of the semester. Each week, the bookstore manager draws a set of 4 numbers or letters from the set of 36. Students whose ID contains the 4 numbers drawn, in any order, may claim a certificate if they stop by the store that week. How many different sets of numbers and letters may be drawn? d. For each set of 4 numbers and letters drawn by the bookstore, say {4, C, K, 7}, what is the largest number of gift certificates that could be claimed? 4. A basketball team has 4 members. a. Ignoring positions played, how many different ways can a group of players be chosen to start in a game? b. How many ways can players be chosen if each is assigned a different position? c. How many ways can the 4 members form a single line for a photograph? d. Before each game, all players names are placed in a hat. One name is drawn to determine who is captain for that game. In 0 games, how many ways can the captain be chosen? 43. A planning committee has 8 members. a. How many ways can a president, vice president, and secretary be selected? b. How many different ways can a 3-person budget subcommittee be selected? c. How many ways can the planning committee line up to use the coffee machine? d. Before each meeting, the administrative assistant calls one of the members of the committee at random and asks this person to pick up the vegetable snack tray for the next meeting. How many ways can this obligation be carried out over meetings? 44. A class of 8 second-graders lines up for a fire drill. a. How many ways can the students pair up? b. How many ways can the 9 pairs of students line up? c. How many ways can the students line up single-file? 4. Special-order license plates are available in most states. a. How many 6-letter words can be made without repetition of letters? (There are 6 letters in the alphabet.) b. How many 6-letter words can be made with repetition? Blue numbers are core exercises.

0838 ch0_p639-693 0//007 0:3 PM Page 66 66 CHAPTER 0 COUNTING AND PROBABILITY 46. a. The Hawaiian alphabet has letters. How many -letter words can be made with repetition of letters? b. How many -letter words can be made without repetition of letters? c. How many groups of different letters can be drawn? 47. A child s birthday wish list has 4 toys, family outings, and 3 party locations. a. How many different celebrations are possible if the parents choose toy or outing or party? b. How many different celebrations are possible if the parents choose toys, outing, and party? 48. A family s holiday wish list has 3 car trips, 4 family gatherings, and 4 performing arts events. a. How many different holiday celebrations are possible if the family does it all? b. How many different holiday celebrations are possible if the family does car trip, family gathering, and performing arts event? c. How many different holiday celebrations are possible if the family does car trip or family gathering or performing arts event? 49. An interview committee to select a college dean is to be chosen from 3 administrators, deans, and 0 faculty leaders. a. How many different 6-member committees can be chosen? b. How many different committees can be chosen if administrator, 3 deans, and faculty leaders must participate? 0. An interview committee to select a college president is to be chosen from 9 board of education members, 3 vice presidents, deans, faculty leaders, and students. a. How many different -member committees can be chosen? b. How many different committees can be chosen if board members and member of each of the other groups are on the committee? c. How many different committees can be chosen if members of each group are on the committee?. Answer yes or no and explain why. a. Does 0 C have the same value as 0 P? b. Does 0 C have the same value as 0 P?. A set has 8 elements. How many subsets are there that contain a. 3 of the elements? b. of the elements? c. Explain why the answers to parts a and b are related. 3. Proofs a. Prove that n P n C. (Hint:! ) b. Prove that n P n. c. Prove that n C n. d. Does n P n n C n? e. Prove that n P n n!. f. Prove that n C n. g. Prove that n P 0 n C 0. h. Prove that n P 0. i. Prove that n C 0. Projects 4. Combinations and Repeated Trials When we flip a coin, we obtain one of two outcomes: H (head) or T (tail). In 4 trials of flipping a coin, there are 6 possible outcomes; see Example 8. a. Using the multiplication principle, show why the 4 trials give 6 outcomes. b. List the 8 different outcomes for flipping a coin 3 times. Group the outcomes by number of heads. c. Evaluate 3 C 0, 3 C, 3 C, 3 C 3. d. If r is the number of tails in n C r in part c, what event is 3C 0 3 C 3 C 3 C 3? e. Why can we use the addition principle in part d? f. What is the sum in part d? g. Using the above results, predict the number of outcomes for flipping a coin times. Check your results by calculating the combinations, C 0, C, C, C 3, C 4, and C and adding them. h. Suggest the meaning of 6 C 0, 6 C, 6 C r, and n C r in terms of coin flipping trials. i. Can your answers to part h be given in terms of either heads or tails?. Functions We can think of the number of permutations as a function. Another way to write the number of permutations is P(n, r). a. P(n, r) has how many input variables? Describe them. b. What is the domain of P(n, r)? c. What is another way to write r 0 and r n? d. What set of numbers contains the range?