Chapter 15

A committee of three is chosen at random from a group of five men and four women. Find the probability that the committee contains two men and one woman. \(\frac{10}{21}\)

\text { A red jack is drawn. } \frac{1}{26}

In how many ways can a sum less than ten be obtained when tossing a pair of dice? 30

Find the probability of each of the following events:A red jack is drawn. \(\frac{1}{26}\)

How many four-person committees can be chosen from five women and three men if each committee must contain at least one man? 65

A committee of four is chosen at random from a group of six men and seven women. Find the probability that the committee contains at least one woman. \(\frac{140}{143}\)

How many different seven-letter permutations can be formed from four identical H's and three identical T's? 35

In how many ways can a sum greater than five be obtained when tossing a pair of dice? 26

A bag contains five red and eight white marbles. Two marbles are drawn in succession with replacement. What is the probability that at least one red marble is drawn? \(\frac{105}{169}\)

How many different eight-letter permutations can be formed from six identical H's and two identical T's? 28

In how many ways can a sum greater than four be obtained when tossing three dice? 212

A bag contains four red, five white, and three blue marbles. Two marbles are drawn in succession with replacement. Find the probability that one red and one blue marble are drawn. \(\frac{1}{6}\)

How many different nine-letter permutations can be formed from three identical A's, four identical B's, and two identical C's? 1260

If no number contains repeated digits, how many numbers greater than 400 can be formed by choosing from the digits \(2,3,4\), and 5 ? [Hint: Consider both three-digit and four-digit numbers.] 36

A bag contains four red and seven blue marbles. Two marbles are drawn in succession without replacement. Find the probability of drawing one red and one blue marble. \(\frac{28}{55}\)

How many different ten-letter permutations can be formed from five identical A's, four identical B's, and one C? 1260

If no number contains repeated digits, how many numbers greater than 5000 can be formed by choosing from the digits \(1,2,3,4,5\), and \(6 ? 1560\)

How many different seven-letter permutations can be formed from the seven letters of the word ALGEBRA? 2520

In how many ways can four boys and three girls be seated in a row of seven seats so that boys and girls occupy alternating seats? 144

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

How many different 11-letter permutations can be formed from the 11 letters of the word MATHEMATICS? 4,989,600

In how many ways can three different mathematics books and four different history books be exhibited on a shelf so that all of the books in a subject area are side by side? \(\quad 288\)

Suppose that a committee of two boys is to be chosen at random from the five boys Al, Bill, Carl, Dan, and Eli. Find the probability of each of the following events:Dan and Eli are both on the committee. \(\frac{1}{10}\)

The probability that a customer in a department store will buy a blouse is \(0.15\), the probability that she will buy a pair of shoes is \(0.10\), and the probability that she will buy both a blouse and a pair of shoes is \(0.05\). Find the probability that the customer will buy a blouse, given that she has already purchased a pair of shoes. Also find the probability that she will buy a pair of shoes, given that she has already purchased a blouse.

In how many ways can \(x^{4} y^{2}\) be written without using exponents? [Hint: One way is xxxxyy.] 15

In how many ways can a true-false test of ten questions be answered? 1024

Solve each problem.Three coins are tossed. Find the probability of getting at least two heads or exactly one tail. \(\frac{1}{2}\)

Suppose that a committee of two boys is to be chosen at random from the five boys Al, Bill, Carl, Dan, and Eli. Find the probability of each of the following events:Bill and Carl are not both on the committee. \(\frac{9}{10}\)

A survey of 500 employees of a company produced the following information.Find the probability that an employee chosen at random (a) is working in a managerial position, given that he or she has a college degree; and (b) has a college degree, given that he or she is working in a managerial position. (a) \(\frac{9}{19}\) (b) \(\frac{9}{10}\)

In how many ways can \(x^{3} y^{4} z^{3}\) be written without using exponents? 4200

If no number contains repeated digits, how many even numbers greater than 3000 can be formed by choosing from the digits \(1,2,3\), and 4 ? 6

Solve each problem.A jar contains seven white, six blue, and ten red marbles. If one marble is drawn at random from the jar, find the probability that (a) the marble is white or blue; (b) the marble is white or red; (c) the marble is blue or red. (a) \(\frac{13}{23}\) (b) \(\frac{17}{23}\) (c) \(\frac{16}{23}\)

Suppose that a committee of two boys is to be chosen at random from the five boys Al, Bill, Carl, Dan, and Eli. Find the probability of each of the following events:Dan or Eli, but not both of them, is on the committee. \(\frac{3}{5}\)

From a survey of 1000 college students, it was found that 450 of them owned cars, 700 of them owned sound systems, and 200 of them owned both a car and a sound system. If a student is chosen at random from the 1000 students, find the probability that the student (a) owns a car, given the fact that he or she owns a sound system, and (b) owns a sound system, given the fact that he or she owns a car. (a) \(\frac{2}{7}\) (b) \(\frac{4}{9}\)

Ten basketball players are gomg to be divided into two teams of five players each for a game. In how many ways can this be done?

Solve each problem.A coin and a die are tossed. Find the probability of getting a head on the coin or a 2 on the die. \(\frac{7}{12}\)

Suppose that a five-person committee is selected at random from the eight people \(\mathrm{Al}\), Barb, Chad, Dominique, Eric, Fern, George, and Harriet. Find the probability of each of the following events:\(\mathrm{Al}\) and Barb are both on the committee. \(\frac{5}{14}\)

Ten basketball players are going to be divided into two teams of five players each for a game. In how many ways can this be done? 126

If no number contains repeated digits, how many odd numbers greater than 40,000 can be formed by choosing from the digits \(1,2,3,4\), and 5 ?

Expand each binomial and simplify.\((x+2 y)^{5}\)

Ten basketball players are going to be divided into two teams of five in such a way that the two best players are on opposite teams. In how many ways can this be done?

In how many ways can Al, Bob, Carol, Don, Ed, Faye, and George be seated in a row of seven seats so that \(\mathrm{Al}\), Bob, and Carol occupy consecutive seats in some order?

Solve each problem.A card is randomly drawn from a deck of 52 playing cards. Find the probability that it is a red card or a face card. (Jacks, queens, and kings are the face cards.) \(\frac{8}{13}\)

Suppose that a five-person committee is selected at random from the eight people \(\mathrm{Al}\), Barb, Chad, Dominique, Eric, Fern, George, and Harriet. Find the probability of each of the following events:George is not on the committee. \(\frac{3}{8}\)

Expand each binomial and simplify.\((x-y)^{8}\)

A box contains nine good light bulbs and four defective bulbs. How many samples of three bulbs contain one defective bulb? How many samples of three bulbs contain at least one defective bulb? \(144 ; 202\)

The license plates for a certain state consist of two letters followed by a four-digit number such that the first digit of the number is not zero. An example is PK-2446. (a) How many different license plates can be produced? 6,084,000 (b) How many different plates do not have a repeated letter? \(\quad 5,850,000\) (c) How many plates do not have any repeated digits in the number part of the plate? \(3,066,336\) (d) How many plates do not have a repeated letter and also do not have any repeated digits? \(\quad 2,948,400\)

Suppose that a five-person committee is selected at random from the eight people \(\mathrm{Al}\), Barb, Chad, Dominique, Eric, Fern, George, and Harriet. Find the probability of each of the following events:Either Chad or Dominique, but not both, is on the committee. \(\frac{15}{28}\)

Expand each binomial and simplify.\(\left(a^{2}-3 b^{3}\right)^{4}\)

How many five-person committees consisting of two juniors and three seniors can be formed from a group of six juniors and eight seniors? 840