Chapter 13

55–75 Solve the problem using the appropriate counting principle(s). Casting a Play A group of 22 aspiring thespians contains ten men and twelve 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?

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.

55–75 Solve the problem using the appropriate counting principle(s). Hockey Lineup A hockey team has 20 players of which twelve 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?

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.

55–75 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?

55–75 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.

55–75 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?

55–75 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.

55–75 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.

55–75 Solve the problem using the appropriate counting principle(s). Selecting Prizewinners From a group of 30 contestants, six are to be chosen as semifinalists, then two 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?

55–75 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?

55–75 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?

55–75 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?

55–75 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. 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 $$C(n, r) \cdot C(n-r, k)=C(n, k) \cdot C(n-k, r)$$

Why is \(\left(\begin{array}{c}{n} \\ {r}\end{array}\right)\) 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 $$\begin{aligned}(x+y)^{2} &=(x+y)(x+y) \\ &=(x+y) x+(x+y) y \\ &=x x+x y+y x+y y \end{aligned}$$ $$\begin{aligned}(x+y)^{3}=&(x+y)(x x+x y+y x+y y) \\\=& x x+x x y+x y x+x y y+y x x \\ &+y x y+y y x+y y y \end{aligned}$$ (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 two \(x^{\prime}\) 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 \(\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)\) in the Binomial Theorem is the same as \(C(n, r) .\)