Chapter 15

How would you explain the fundamental principle of counting to a friend who missed class the day it was discussed?

Expand each binomial and simplify.\(\left(x+\frac{1}{n}\right)^{6}\)

Give two or three simple illustrations of the fundamental principle of counting.

In a recent election there were 1000 eligible voters. They were asked to vote on two issues, \(A\) and \(B\). The results were as follows: 300 people voted for \(A, 400\) people voted for \(B\), and 175 voted for both \(A\) and \(B\). If one person is chosen at random from the 1000 eligible voters, find the probability that the person voted for A or B. \(0.525\)

Suppose that a box of ten items from a manufacturing company is known to contain two defective and eight nondefective items. A sample of three items is selected at random. Find the probability of each of the following events:The sample contains all nondefective items. \(\frac{7}{15}\)

In how many ways can six people be divided into two groups so that there are four in one group and two in the other? In how many ways can six people be divided into two groups of three each? \(15 ; 20\)

Expand each binomial and simplify.\((1-\sqrt{2})^{5}\)

How many five-element subsets containing \(A\) and \(B\) can be formed from the set \(\\{A, B, C, D, E, F, G, H\\}\) ? 20

A company has 500 employees among whom 200 are females, 15 are high-level executives, and 7 of the high-level executives are females. If one of the 500 employees is chosen at random, find the probability that the person chosen is female or is a high-level executive. \(0.416\)

Suppose that a box of ten items from a manufacturing company is known to contain two defective and eight nondefective items. A sample of three items is selected at random. Find the probability of each of the following events:The sample contains one defective and two nondefective items. \(\frac{7}{15}\)

Expand each binomial and simplify.\((-a+b)^{3}\)

A bag contains one red and two white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:One marble drawn is red, and one marble drawn is white. \(\frac{2}{3}\)

How many four-element subsets containing A or B but not both \(A\) and \(B\) can be formed from the set \(\\{A, B, C\), D, E, F, G\\}? 20

Suppose that a box of ten items from a manufacturing company is known to contain two defective and eight nondefective items. A sample of three items is selected at random. Find the probability of each of the following events:The sample contains two defective and one nondefective item. \(\frac{1}{15}\)

A bag contains one red and two white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:The first marble drawn is red and the second is white. \(\frac{1}{3}\)

How many different five-person committees can be selected from nine people if two of those people refuse to serve together on a committee? 91

A bag contains one red and two white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are white. \(\frac{1}{3}\)

Bill, Carol, and Alice are to be seated at random in a row of three seats. Find the probability that Bill and Carol will be seated side by side. \(\frac{2}{3}\)

How many different line segments are determined by five points? By six points? By seven points? By \(n\) points? \(10 ; 15 ; 21 ; \frac{n(n-1)}{2}\)

Two dice are tossed 720 times. How many times would you expect to get a sum greater than 9 ? 120

A bag contains one red and two white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are red. 0

April, Bill, Carl, and Denise are to be seated at random in a row of four chairs. What is the probability that April and Bill will occupy the end seats? \(\frac{1}{6}\)

(a) How many five-card hands consisting of two kings (c) How many five-card hands consisting of three cards and three aces can be dealt from a deck of 52 playof one face value and two cards of another face ing cards? value can be dealt from a deck of 52 playing cards? (b) How many five-card hands consisting of three 3744 kings and two aces can be dealt from a deck of 52 playing cards? 24

Four coins are tossed 80 times. How many times would you expect to get one head and three tails? 20

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are red. \(\frac{5}{68}\)

Three boys and two girls are to be seated at random in a row of five seats. What is the probability that the boys and girls will be in alternating seats? \(\frac{1}{10}\)

Your friend is having difficulty distinguishing between permutations and combinations in problem-solving situations. What might you do to help her?

Four coins are tossed 144 times. How many times would you expect to get four tails? 9

A committee of four girls is to be chosen at random from the five girls Alice, Becky, Candy, Dee, and Elaine. Find the probability that Elaine is not on the committee. \(\frac{1}{5}\)

Explain the difference between a permutation and a combination. Give an example of each one to illustrate your explanation.

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are white. \(\frac{33}{68}\)

Three boys and two girls are to be seated at random in a row of five seats. What is the probability that the boys and girls will be in alternating seats? \(\frac{1}{10}\)

Your friend is having difficulty distinguishing between permutations and combinations in problem-solving situations. What might you do to help her?

Two dice are tossed 300 times. How many times would you expect to get a double? 50

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:One red and one white marble are drawn. \(\frac{15}{34}\)

Four different mathematics books and five different history books are randomly placed on a shelf. What is the probability that all of the books on a subject are side by side? \(\frac{1}{63}\)

In how many ways can six people be seated at a circular table? [Hint: Moving each person one place to the right (or left) does not create a new seating.] 120

Three coins are tossed 448 times. How many times would you expect to get three heads? 56

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:At least one marble drawn is red. \(\frac{35}{68}\)

Suppose 5000 tickets are sold in a lottery. There are three prizes: The first is \(\$ 1000\), the second is \(\$ 500\), and the third is \(\$ 100\). What is the mathematical expectation of winning?

63 Each of three letters is to be mailed in any one of five different mailboxes. What is the probability that all will be mailed in the same mailbox?

The quantity \(P(8,3)\) can be expressed completely in factorial notation as follows: $$ P(8,3)=\frac{P(8,3) \cdot 5 !}{5 !}=\frac{(8 \cdot 7 \cdot 6)(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}{5 !}=\frac{8 !}{5 !} $$ Express each of the following in terms of factorial notation. (a) \(P(7,3)\) See below (b) \(P(9,2)\) See below (c) \(P(10,7)\) See below (d) \(P(n, r), \quad r \leq n\) and 0 ! is defined to be \(1 \frac{n !}{(n-r) !}\)

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are white. \(\frac{1}{12}\)

Randomly form a four-digit number by using the digits \(2,3,4\), and 6 once each. What is the probability that the number formed is greater than \(4000 ? \quad \frac{1}{2}\)

Your friend challenges you with the following game: You are to roll a pair of dice, and he will give you \(\$ 5\) if you roll a sum of 2 or \(12, \$ 2\) if you roll a sum of 3 or 11 , \(\$ 1\) if you roll a sum of 4 or 10 . Otherwise you are to pay him \(\$ 1\). Should you play the game? It is a fair game.

Randomly select one of the 120 permutations of the letters \(a, b, c, d\), and \(e\). Find the probability that in the chosen permutation, the letter \(a\) precedes the \(b\) (the \(a\) is to the left of the \(b\) ). \(\frac{1}{2}\)

Compute \(C(7,3)\) and \(C(7,4)\). Compute \(C(8,2)\) and \(C(8,6)\). Compute \(C(9,8)\) and \(C(9,1)\). Now argue that \(C(n, r)=C(n, n-r)\) for \(r \leq n\).

A contractor bids on a building project. There is a probability of \(0.8\) that he can show a profit of \(\$ 30,000\) and a probability of \(0.2\) that he will have to absorb a loss of \(\$ 10,000\). What is his mathematical expectation? \(\$ 22,000\)

A committee of four is chosen at random from a group of six women and five men. Find the probability that the committee contains two women and two men.

Suppose a person tosses two coins and receives \(\$ 5\) if 2 heads come up, receives \(\$ 2\) if 1 head and 1 tail come up, and has to pay \(\$ 2\) if 2 tails come up. Is it a fair game for him? Yes