Chapter 13

19–32 These problems involve permutations. Seating Arrangements In how many different ways can six people be seated in a row of six chairs?

A combination lock has 60 different positions. To open the lock, the dial is turned to a certain number in the clockwise direction, then to a number in the counterclockwise direction, and finally to a third number in the clockwise direction. If successive numbers in the combination cannot be the same, how many different combinations are possible?

The president of a large company selects six employees to receive a special bonus. He claims that the six employees are chosen randomly from among the 30 employees, of which 19 are women and 11 are men. What is the probability that no woman is chosen?

Defective Light Bulbs The DimBulb Lighting Company manufactures light bulbs for appliances such as ovens and refrigerators. Typically, 0.5\(\%\) of their bulbs are defective. From a crate with 100 bulbs, 3 are tested. Find the probability that the given event occurs. (a) All 3 bulbs are defective. (b) One or more bulbs is defective.

19–32 These problems involve permutations. Three-Letter Words How many three-letter “words” can be made from the letters FGHIJK? (Letters may not be repeated.)

A true-false test contains ten questions. In how many different ways can this test be completed?

An exam has ten true-false questions. A student who has not studied answers all ten questions by just guessing. Find the probability that the student correctly answers the given number of questions. (a) All ten questions (b) Exactly seven questions

Quality Control An assembly line that manufactures fuses for automotive use is checked every hour to ensure the quality of the finished product. Ten fuses are selected randomly, and if any one of the ten is found to be defective, the process is halted and the machines are recalibrated. Suppose that at a certain time 5\(\%\) of the fuses being produced are actually defective. What is the probability the assembly line is halted at that hour's quality check?

19–32 These problems involve permutations. Letter Permutations How many permutations are possible from the letters of the word LOVE?

An automobile dealer offers five models. Each model comes in a choice of four colors, three types of stereo equipment, with or without air conditioning, and with or without a sunroof. In how many different ways can a customer order an auto from this dealer?

To control the quality of their product, the Bright-Light Company inspects three light bulbs out of each batch of ten bulbs manufactured. If a defective bulb is found, the batch is discarded. Suppose a batch contains two defective bulbs. What is the probability that the batch will be discarded?

Sick Leave The probability that a given worker at the Dyno Nutrition will call in sick on a Monday is \(0.04 .\) The packaging department has 8 workers. What is the probability that 2 or more packaging workers will call in sick next Monday?

19–32 These problems involve permutations. Three-Digit Numbers How many different three-digit whole numbers can be formed using the digits 1, 3, 5, and 7 if no repetition of digits is allowed?

Political Surveys In a certain county, 60\(\%\) of the voters are in favor of an upcoming school bond initiative. If 5 voters are interviewed at random, what is the probability that exactly 3 of them will favor the initiative?

19–32 These problems involve permutations. Piano Recital A pianist plans to play eight pieces at a recital. In how many ways can she arrange these pieces in the program?

How many monograms consisting of three initials are possible?

Making Words A monkey is trained to arrange wooden blocks in a straight line. He is then given six blocks showing the letters \(A, E, H, L, M, T .\) What is the probability that he will arrange them to spell the word HAMLET?

Pharmaceuticals \(\quad\) A drug used to prevent motion sickness is found to be effective about 75\(\%\) of the time. Six friends, prone to seasickness, go on a sailing cruise, and all take the drug. Find the probability of each event. (a) None of the friends gets seasick. (b) All of the friends get seasick. (c) Exactly 3 get seasick (d) At least 2 get seasick.

19–32 These problems involve permutations. Running a Race In how many different ways can a race with nine runners be completed, assuming there is no tie?

A state has registered 8 million automobiles. To simplify the license plate system, a state employee suggests that each plate display only two letters followed by three digits. Will this system create enough different license plates for all the vehicles registered?

A monkey is trained to arrange wooden blocks in a straight line. He is then given 11 blocks showing the letters \(A, B, B, I, I, L, O, P, R, T, Y .\) What is the probability that the monkey will arrange the blocks to spell the word PROBABILITY?

Reliability of a Machine A machine used in a manufacturing process has 4 separate components, each of which has a 0.01 probability of failing on any given day. If any component fails, the entire machine breaks down. Find the probability that on a given day the indicated event occurs. (a) The machine breaks down. (b) The machine does not break down. (c) Only one component does not fail.

19–32 These problems involve permutations. Signal Flags A ship carries five signal flags of different colors. How many different signals can be sent by hoisting exactly three of the five flags on the ship’s flagpole in different orders?

A state license plate design has six places. Each plate begins with a fixed number of letters, and the remaining places are filled with digits. (For example, one letter followed by five digits, two letters followed by four digits, and so on.) The state has 17 million registered vehicles. (a) The state decides to change to a system consisting of one letter followed by five digits. Will this design allow for enough different plates to accommodate all the vehicles registered? (b) Find a system that will be sufficient if the smallest possible number of letters is to be used.

Eight horses are entered in a race.You randomly predict a particular order for the horses to complete the race. What is the probability that your prediction is correct?

Genetics Huntington's disease is a hereditary ailment caused by a recessive gene. If both parents carry the gene but do not have the disease, there is a 0.25 probability that an offspring will fall victim to the condition. A newly wed couple find through genetic testing that they both carry the gene (but do not have the disease). If they intend to have four children, find the probability of each event. (a) At least one child gets the disease. (b) At least 3 of the children get the disease.

19–32 These problems involve permutations. Contest Prizes In how many ways can first, second, and third prizes be awarded in a contest with 1000 contestants?

In how many ways can a president, vice president, and secretary be chosen from a class of 30 students?

Many genetic traits are controlled by two genes, one dominant and one recessive. In Gregor Mendel’s original experiments with peas, the genes controlling the height of the plant are denoted by T (tall) and t (short). The gene T is dominant, so a plant with the genotype (genetic makeup) TT or Tt is tall, whereas one with genotype tt is short. By a statistical analysis of the offspring in his experiments, Mendel concluded that offspring inherit one gene from each parent, and each possible combination of the two genes is equally likely. If each parent has the genotype Tt, then the following chart gives the possible genotypes of the offspring: (Table not Copy) Find the probability that a given offspring of these parents will be (a) tall or (b) short.

Selecting Cards Three cards are randomly selected from a standard 52 -card deck, one at a time, with each card re- placed in the deck before the next one is picked. Find the probability of each event. (a) All three cards are hearts. (b) Exactly two of the cards are spades. (d) None of the cards is a diamond. (d) At least one of the cards is a club.

19–32 These problems involve permutations. Class Officers In how many ways can a president, vice president, secretary, and treasurer be chosen from a class of 30 students?

In how many ways can a president, vice president, and secretary be chosen from a class of 20 females and 30 males if the president must be a female and the vice president a male?

19–32 These problems involve permutations. Seating Arrangements In how many ways can five students be seated in a row of five chairs if Jack insists on sitting in the first chair?

A senate subcommittee consists of ten Democrats and seven Republicans. In how many ways can a chairman, vice chairman, and secretary be chosen if the chairman must be a Democrat and the vice chairman must be a Republican?

Determine whether the events \(E\) and \(F\) in the given experiment are mutually exclusive. The experiment consists of selecting a person at random. (a) \(E :\) The person is male \(F:\) The person is female (b) \(E :\) The person is tall \(F:\) The person is blond

Telephone Marketing A mortgage company advertises its rates by making unsolicited telephone calls to random numbers. About 2\(\%\) of the calls reach consumers who are interested in the company's services. A telephone consultant can make 100 calls per evening shift. (a) What is the probability that 2 or more calls will reach an interested party in one shift? (b) How many calls does a consultant need to make to ensure at least a 0.5 probability of reaching one or more interested parties? [Hint: Use trial and error.]

Determine whether the events \(E\) and \(F\) in the given experiment are mutually exclusive. The experiment consists of choosing at random a student from your class. (a) \(E :\) The student is female \(F :\) The student wears glasses (b) \(E :\) The student has long hair \(F :\) The student is male

Social Security numbers consist of nine digits, with the first digit between 0 and 6, inclusive. How many Social Security numbers are possible?

33–40 These problems involve distinguishable permutations. Arrangements In how many ways can two blue marbles and four red marbles be arranged in a row?

Five-letter “words” are formed using the letters A, B, C, D, E, F, G. How many such words are possible for each of the following conditions? (a) No condition is imposed. (b) No letter can be repeated in a word. (c) Each word must begin with the letter A. (d) The letter C must be in the middle. (e) The middle letter must be a vowel.

A die is rolled and the number showing is observed. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The number is even \(F:\) The number is odd (b) \(E :\) The number is even \(F:\) The number is greater than 4

33–40 These problems involve distinguishable permutations. Arrangements In how many different ways can five red balls, two white balls, and seven blue balls be arranged in a row?

How many five-letter palindromes are possible? (A palindrome is a string of letters that reads the same backward and forward, such as the string XCZCX.)

A die is rolled and the number showing is observed. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The number is greater than 3 \(F :\) The number is less than 5 (b) \(E :\) The number is divisible by 3 \(F :\) The number is less than 3

33–40 These problems involve distinguishable permutations. Arranging Coins In how many different ways can four pennies, three nickels, two dimes, and three quarters be arranged in a row?

A certain computer programming language allows names of variables to consist of two characters, the first being any letter and the second any letter or digit. How many names of variables are possible?

A card is drawn at random from a standard 52 -card deck. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The card is a face card \(F :\) The card is a spade (b) \(E :\) The card is a heart \(F :\) The card is a spade

33–40 These problems involve distinguishable permutations. Arranging Letters In how many different ways can the letters of the word ELEEMOSYNARY be arranged?

A card is drawn at random from a standard 52 -card deck. Determine whether the events \(E\) and \(F\) are mutually exclusive. Then find the probability of the event \(E \cup F .\) (a) \(E :\) The card is a club \(F :\) The card is a king (b) \(E :\) The card is an ace \(F :\) The card is a spade

33–40 These problems involve distinguishable permutations. Distributions A man bought three vanilla ice-cream cones, two chocolate cones, four strawberry cones, and five butterscotch cones for his 14 children. In how many ways can he distribute the cones among his children?