Chapter 13

In how many ways can four men and four women be seated in a row of eight seats for the following situations? (a) The women are to be seated together, and the men are to be seated together. (b) They are to be seated alternately by gender.

33–40 These problems involve distinguishable permutations. Room Assignments When seven students take a trip, they find a hotel with three rooms available—a room for one person, a room for two people, and a room for three people. In how many different ways can the students be assigned to these rooms? (One student has to sleep in the car.)

In how many ways can five different mathematics books be placed on a shelf if the two algebra books are to be placed next to each other?

33–40 These problems involve distinguishable permutations. Work Assignments Eight workers are cleaning a large house. Five are needed to clean windows, two to clean the carpets, and one to clean the rest of the house. In how many different ways can these tasks be assigned to the eight workers?

Eight mathematics books and three chemistry books are to be placed on a shelf. In how many ways can this be done if the mathematics books are next to each other and the chemistry books are next to each other?

An American roulette wheel has 38 slots: two of the slots are numbered 0 and 00, and the rest are numbered from 1 to 36. Find the probability that the ball lands in an odd-numbered slot or in a slot with a number higher than 31.

33–40 These problems involve distinguishable permutations. Jogging Routes A jogger jogs every morning to his health club, which is eight blocks east and five blocks north of his home. He always takes a route that is as short as possible, but he likes to vary it (see the figure). How many different routes can he take? [Hint: The route shown can be thought of as \(ENNEEENENEENE\), where \(E\) is East and \(N\) is North.]

Three-digit numbers are formed using the digits 2, 4, 5, and 7, with repetition of digits allowed. How many such numbers can be formed if (a) the numbers are less than 700? (b) the numbers are even? (c) the numbers are divisible by 5?

A toddler has eight wooden blocks showing the letters \(A, E, I, G, L, N, T,\) and \(R .\) What is the probability that the child will arrange the letters to spell one of the words TRIANGLE or INTEGRAL?

41–54 These problems involve combinations. Choosing Books In how many ways can three books be chosen from a group of six?

A committee of five is chosen randomly from a group of six males and eight females. What is the probability that the committee includes either all males or all females?

41–54 These problems involve combinations. Pizza Toppings In how many ways can three pizza toppings be chosen from 12 available toppings?

Explain why in any group of 677 people, at least two people must have the same pair of initials.

In the 6/49 lottery game a player selects six numbers from 1 to 49. What is the probability of selecting at least five of the six winning numbers?

41–54 These problems involve combinations. Choosing a Group In how many ways can six people be chosen from a group of ten?

A jar contains six red marbles numbered 1 to 6 and ten blue marbles numbered 1 to 10. A marble is drawn at random from the jar. Find the probability that the given event occurs. (a) The marble is red. (b) The marble is odd-numbered. (c) The marble is red or odd-numbered. (d) The marble is blue or even-numbered.

41–54 These problems involve combinations. Committee In how many ways can a committee of three members be chosen from a club of 25 members?

A coin is tossed twice. Let \(E\) and \(F\) be the following events: $$\begin{array}{l}{E : \text { The first toss shows heads }} \\ {F : \text { The second toss shows heads }}\end{array}$$ (a) Are the events \(E\) and \(F\) independent? (b) Find the probability of showing heads on both tosses.

41–54 These problems involve combinations. Draw Poker Hands How many different five-card hands can be dealt from a deck of 52 cards?

A die is rolled twice. Let \(E\) and \(F\) be the following events: $$\begin{array}{l}{\text { E: The first roll shows a six }} \\ {F : \text {The second roll shows a six }}\end{array}$$ (a) Are the events \(E\) and \(F\) independent? (b) Find the probability of showing a six on both rolls.

41–54 These problems involve combinations. Stud Poker Hands How many different seven-card hands can be picked from a deck of 52 cards?

41–54 These problems involve combinations. Choosing Exam Questions A student must answer seven of the ten questions on an exam. In how many ways can she choose the seven questions?

41–54 These problems involve combinations. Three-Topping Pizzas A pizza parlor offers a choice of 16 different toppings. How many three-topping pizzas are possible?

A die is rolled twice. What is the probability of showing a one on both rolls?

41–54 These problems involve combinations. Violin Recital A violinist has practiced 12 pieces. In how many ways can he choose eight of these pieces for a recital?

A die is rolled twice. What is the probability of showing a one on the first roll and an even number on the second roll?

41–54 These problems involve combinations. Choosing Clothing If a woman has eight skirts, in how many ways can she choose five of these to take on a weekend trip?

A card is drawn from a deck and replaced, and then a second card is drawn. (a) What is the probability that both cards are aces? (b) What is the probability that the first is an ace and the second a spade?

41–54 These problems involve combinations. Field Trip In how many ways can seven students from a class of 30 be chosen for a field trip?

A roulette wheel has 38 slots: Two slots are numbered 0 and 00, and the rest are numbered 1 to 36. A player places a bet on a number between 1 and 36 and wins if a ball thrown into the spinning roulette wheel lands in the slot with the same number. Find the probability of winning on two consecutive spins of the roulette wheel.

A researcher claims that she has taught a monkey to spell the word MONKEY using the five wooden letters \(E, O, K, M, N, Y .\) If the monkey has not actually learned anything and is merely arranging the blocks randomly, what is the probability that he will spell the word correctly three consecutive times?

What is the probability of rolling “snake eyes” (double ones) three times in a row with a pair of dice?

41–54 These problems involve combinations. Lottery In the 6/49 lottery game, a player picks six numbers from 1 to 49. How many different choices does the player have?

In the 6/49 lottery game, a player selects six numbers from 1 to 49 and wins if he selects the winning six numbers. What is the probability of winning the lottery two times in a row?

55–75 Solve the problem using the appropriate counting principle(s). Lottery In the California Lotto game, a player chooses six numbers from 1 to 53. It costs $1 to play this game. How much would it cost to buy every possible combination of six numbers to ensure picking the winning six numbers?

Jar A contains three red balls and four white balls. Jar B contains five red balls and two white balls. Which one of the following ways of randomly selecting balls gives the greatest probability of drawing two red balls? (i) Draw two balls from jar B. (ii) Draw one ball from each jar. (iii) Put all the balls in one jar, and then draw two balls.

55–75 Solve the problem using the appropriate counting principle(s). Choosing a Committee A class has 20 students, of which 12 are females and 8 are males. In how many ways can a committee of five students be picked from this class under each condition? (a) No restriction is placed on the number of males or females on the committee. (b) No males are to be included on the committee. (c) The committee must have three females and two males.

A slot machine has three wheels: Each wheel has 11 positions—a bar and the digits 0, 1, 2, . . . , 9. When the handle is pulled, the three wheels spin independently before coming to rest. Find the probability that the wheels stop on the following positions. (a) Three bars (b) The same number on each wheel (c) At least one bar

55–75 Solve the problem using the appropriate counting principle(s). Subsets A set has eight elements. (a) How many subsets containing five elements does this set have? (b) How many subsets does this set have?

Find the probability that in a group of eight students at least two people have the same birthday.

55–75 Solve the problem using the appropriate counting principle(s). Travel Brochures A travel agency has limited numbers of eight different free brochures about Australia. The agent tells you to take any that you like, but no more than one of any kind. How many different ways can you choose brochures (including not choosing any)?

What is the probability that in a group of six students at least two have birthdays in the same month?

55–75 Solve the problem using the appropriate counting principle(s). Hamburgers A hamburger chain gives their customers a choice of ten different hamburger toppings. In how many different ways can a customer order a hamburger?

A student has locked her locker with a combination lock, showing numbers from 1 to 40, but she has forgotten the three-number combination that opens the lock. In order to open the lock, she decides to try all possible combinations. If she can try ten different combinations every minute, what is the probability that she will open the lock within one hour?

55–75 Solve the problem using the appropriate counting principle(s). To Shop or Not to Shop Each of 20 shoppers in a shopping mall chooses to enter or not to enter the Dressfastic clothing store. How many different outcomes of their decisions are possible?

A mathematics department consists of ten men and eight women. Six mathematics faculty members are to be selected at random for the curriculum committee. (a) What is the probability that two women and four men are selected? (b) What is the probability that two or fewer women are selected? (c) What is the probability that more than two women are selected?

55–75 Solve the problem using the appropriate counting principle(s). Doubles Tennis From a group of ten male and ten female tennis players, two men and two women are to face each other in a men-versus-women doubles match. In how many different ways can this match be arranged?

Twenty students are arranged randomly in a row for a class picture. Paul wants to stand next to Phyllis. Find the probability that he gets his wish.

55–75 Solve the problem using the appropriate counting principle(s). Dance Committee A school dance committee is to consist of two freshmen, three sophomores, four juniors, and five seniors. If six freshmen, eight sophomores, twelve juniors, and ten seniors are eligible to be on the committee, in how many ways can the committee be chosen?

Eight boys and 12 girls are arranged in a row. What is the probability that all the boys will be standing at one end of the row and all the girls at the other end?