Chapter 14

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 is a spade?

Roulette 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.

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?

Making Words \(\quad\) 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?

Solve the problem using the appropriate counting principle(s). Choosing a Committee A class has 20 students, of whom 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.

Snake Eyes What is the probability of rolling "snake eyes" (double ones) three times in a row with a pair of dice? (PICTURE NOT COPY)

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?

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

Solve the problem using the appropriate counting principle(s). Choosing a Committee A committee of six is to be chosen from a class of 20 students. The committee is to consist of a president, a vice president, and four other members. In how many different ways can the committee be picked?

Solve the problem using the appropriate counting principle(s). Choosing a Group Sixteen boys and nine girls go on a camping trip. In how many ways can a group of six be selected to gather firewood, given the following conditions? (a) The group consists of two girls and four boys. (b) The group contains at least two girls.

Slot Machine A slot machine has three wheels. Each wheel has 11 positions: a bar and the digits \(0,1,2, \ldots, 9\) 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

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?

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

Solve the problem using the appropriate counting principle(s). Casting a Play A group of 22 aspiring thespians contains 10 men and 12 women. For the next play the director wants to choose a leading man, a leading lady, a supporting male role, a supporting female role, and eight extras- three women and five men. In how many ways can the cast be chosen?

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

Solve the problem using the appropriate counting principle(s). Hockey Lineup A hockey team has 20 players, of whom 12 play forward, six play defense, and two are goalies. In how many ways can the coach pick a starting lineup consisting of three forwards, two defense players, and one goalie?

Combination Lock 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. 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?

Solve the problem using the appropriate counting principle(s). Choosing a Pizza A pizza parlor offers four sizes of pizza (small, medium, large, and colossus), two types of crust (thick and thin), and 14 different toppings. How many different pizzas can be made with these choices?

Committee Membership 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?

Solve the problem using the appropriate counting principle(s). Arranging a Class Picture In how many ways can ten students be arranged in a row for a class picture if John and Jane want to stand next to each other and Mike and Molly also insist on standing next to each other?

Class Photo 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?

Solve the problem using the appropriate counting principle(s). Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for each of the following arrangements? (a) The first seat is to be occupied by a man. (b) The first and last seats are to be occupied by women.

The "Second Son" Paradox Mrs. Smith says, "I have two children. The older one is named William." "Mrs. Jones replies, "One of my two children is also named William." For each woman, list the sample space for the genders of her children, and calculate the probability that her other child is also a son. Explain why these two probabilities are different.

Solve the problem using the appropriate counting principle(s). Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for each of the following arrangements? (a) The women are to be seated together. (b) The men and women are to be seated alternately by gender.

The "Oldest Son or Daughter" Phenomenon Poll your class to determine how many of your male classmates are the oldest sons in their families and how many of your female classmates are the oldest daughters in their families. You will most likely find that they form a majority of the class. Explain why a randomly selected individual has a high probability of being the oldest son or daughter in his or her family.

Solve the problem using the appropriate counting principle(s). Selecting Prizewinners From a group of 30 contestants, 6 are to be chosen as semifinalists, then 2 of those are chosen as finalists, and then the top prize is awarded to one of the finalists. In how many ways can these choices be made in sequence?

Solve the problem using the appropriate counting principle(s). Choosing a Delegation Three delegates are to be chosen from a group of four lawyers, a priest, and three professors. In how many ways can the delegation be chosen if it must include at least one professor?

Solve the problem using the appropriate counting principle(s). Choosing a Committee In how many ways can a committee of four be chosen from a group of ten if two people refuse to serve together on the same committee?

Solve the problem using the appropriate counting principle(s). Geometry Twelve dots are drawn on a page in such a way that no three are collinear. How many straight lines can be formed by joining the dots?

Solve the problem using the appropriate counting principle(s). Parking Committee A five-person committee consisting of students and teachers is being formed to study the issue of student parking privileges. Of those who have expressed an interest in serving on the committee, 12 are teachers and 14 are students. In how many ways can the committee be formed if at least one student and one teacher must be included?

Complementary Combinations Without performing any calculations, explain in words why the number of ways of choosing two objects from ten objects is the same as the number of ways of choosing eight objects from ten objects. In general, explain why $$ C(n, r)=C(n, n-r) $$

An Identity Involving Combinations Kevin has ten different marbles, and he wants to give three of them to Luke and two to Mark. In how many ways can he choose to do this? There are two ways of analyzing this problem: He could first pick three for Luke and then two for Mark, or he could first pick two for Mark and then three for Luke. Explain how these two viewpoints show that $$ C(10,3) \cdot C(7,2)=C(10,2) \cdot C(8,3) $$ In general, explain why $$ C(n, r) \cdot C(n-r, k)=a(n, k) \cdot C(n-k, r) $$

Why Is \(\left(x^{n}\right)\) the Same as \(C(n, r) ?\) This exercise explains why the binomial coefficients \(\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)\) that appear in the expansion of \((x+y)^{n}\) are the same as \(C(n, r),\) the number of ways of choosing \(r\) objects from \(n\) objects. First, note that expanding a binomial using only the Distributive Property gives $$ \begin{aligned}(x+y)^{2} &=(x+y)(x+y) \\ &=(x+y) x+(x+y) y \\ &=x x+x y+y x+y y \\\\(x+y)^{3} &=(x+y)(x x+x y+y x+y y) \\ &=x x x+x x y+x y x+x y y+y x x \\ &+y x y+y y x+y y y \end{aligned} $$ (a) Expand \((x+y)^{5}\) using only the Distributive Property. (b) Write all the terms that represent \(x^{2} y^{3}\) together. These are all the terms that contain two \(x^{\prime}\) s and three \(y^{\prime} s .\) (c) Note that the two \(x\) 's appear in all possible positions. Conclude that the number of terms that represent \(x^{2} y^{3}\) is \(C(5,2) .\) (d) In general, explain why \((r)\) in the Binomial Theorem is the same as \(C(n, r) .\)