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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

Derive formula (1), using the generalized multiplication principle and the formula for (nr).
1. Calculate the number of ways that 38 students can be assigned to four seminars of size 10, 12, 10, and 6, respectively. 2. Calculate the number of ways that 65 phone numbers can be distributed to 5 campaign workers if each worker gets the same number of names. 3. One octillion is 1028, or 10
1. List all subsets of the set {a, b}. 2. Draw a two-circle Venn diagram, and shade the portion corresponding to the set (S ∪ T')'. 3. There are 16 contestants in a tennis tournament. How many different possibilities are there for the two people who will play in the final round? 4. In how many
Out of a group of 115 applicants for jobs at the World Bank, 70 speak French, 65 speak Spanish, 65 speak German, 45 speak French and Spanish, 35 speak Spanish and German, 40 speak French and German, and 35 speak all three languages. How many of the people speak none of the three languages?
How many members thought that the priority should be clean air only? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean air, 42 for recycling, 13
How many members thought that the priority should be clean water or clean air, but not both? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean air,
How many members thought that the priority should be clean water or recycling but not clean air? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean
How many members thought that the priority should be clean air and recycling but not clean water? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean
How many members thought that the priority should be exactly one of the three issues? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean air, 42 for
How many members thought that recycling should not be a priority? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean air, 42 for recycling, 13 for
How many members thought that the priority should be recycling but not clean air? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean air, 42 for
How many members thought that the priority should be something other than one of these three issues? The 100 members of the Earth Club were asked what they felt the club's priorities should be in the coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for
1. Words How many different nine-letter words (i.e., sequences of letters) can be made by using four Ss and five Ts? 2. Twenty people take an exam. How many different possibilities are there for the set of people who pass the exam? 3. A survey at a small New England college showed that 400 students
1. How many different meals can be chosen if there are 6 appetizers, 10 main dishes, and 8 desserts, assuming that a meal consists of one item from each category? 2. On an essay test, there are five questions worth 20 points each. In how many ways can a student get 10 points on one question, 15
How many strings of length 8 can be formed from the letters A, B, C, D, and E ? How many of the strings have at least one E?
1. How many different five-person basketball teams can be formed from a pool of 12 players? 2. Fourteen students in the 30-student eighth grade are to be chosen to tour the United Nations. How many different groups of 14 are possible?
In one ZIP code, there are 40,000 households. Of them, 4000 households get Fancy Diet Magazine, 10,000 households get Clean Living Journal, and 1500 households get both publications. How many households get neither?
1. At each stage in a decision process, a computer program has three branches. There are 10 stages at which these branches appear. How many different paths could the process follow? 2. Sixty people apply for 10 job openings. In how many ways can all of the jobs be filled? 3. A computerized test
1. In how many ways can a teacher divide a class of 21 students into groups of 7 students each? 2. In how many ways can 14 different candies be distributed to 14 scouts? 3. In how many ways can 20 people be divided into groups of 5 each? 4. An Internet company is considering three candidates for
1. How many diagonals does an n-sided polygon have? 2. Racetracks have a compound bet called the daily double, in which the bettor tries to select the winners of the first two races. If eight horses compete in the first race and six horses compete in the second race, determine the possible number
1. A designer of a window display wants to form a pyramid with 15 hats. She wants to place the five men's hats in the bottom row, the four women's hats in the next row, next the three baseball caps, then the two berets, and a clown's hat at the top. All of the hats are different. How many displays
1. How many hands of five cards contain exactly three aces? 2. Numbers How many three-digit numbers are there in which exactly two digits are alike?
1. Draw a three-circle Venn diagram, and shade the portion corresponding to the set R' ∩ (S ∪ T). 2. Determine the first three terms in the binomial expansion of (x - 2y)12. 3. Balls in an Urn An urn contains 14 numbered balls, of which 8 are red and 6 are green. How many different
1. How many three-digit numbers are there in which no two digits are alike? 2. How many different pitcher-catcher pairs can be formed? 3. In how many ways can the six players be seated in a row if no one sits next to someone who plays the same position? 4. Fraternity and sorority names consist of
1. A family consisting of two parents and four children is to be seated in a row for a picture. How many different arrangements are possible in which the children are seated together? 2. If 10 lines are drawn in the plane so that no two of them are parallel and no three lines intersect at the same
1. A consulting engineer agrees to spend three days at Widgets International, four days at Gadgets Unlimited, and three days at Doodads Incorporated in the next two workweeks. In how many different ways can she schedule her consultations? 2. A set of books can be arranged on a bookshelf in 120
In the United States Senate, each state is represented by one junior senator and one senior senator. In how many ways can a committee of five senators from the 12 Midwest states be represented if (a) The committee must consist of two senior senators and three junior senators? (b) No two senators
1. Suppose that you are voting in an election for state delegate. Two state delegates are to be elected from among seven candidates. In how many different ways can you cast your ballot? Note: You may vote for two candidates. However, some people "single-shoot," and others don't pull any levers.2.
1. The call letters of radio stations in the United States consist of either three or four letters, where the first letter is K or W. How many different call letters are possible? 2. A group of students can be arranged in a row of seats in 479,001,600 ways. How many students are there?
1. There are 25 people in a department who must be deployed to work on three projects requiring 10, 9, and 6 people. All of the people are eligible for all jobs. Calculate the number of ways this can be done. 2. Calculate the number of license plates that can be formed by using three distinct
1. What is the relationship between the two sets A and B if A ∩ B = Ø? 2. What is the relationship between the two sets A and B if n (A ∪ B) = n (A) + n (B)? 3. Suppose that A and B are subsets of the set U. Under what circumstance will A ∩ B = B? 4. Suppose that A and B are subsets of the
1. (True or False) n (A) + n(B) = n (A ∩ B) + n (A ∪ B). 2. The empty set is a subset of every set. 3. Explain why n # (n - 1)! = n!. 4. Use the result in Exercise 73 to explain why 0! is defined to be 1.
Express in your own words the difference between a permutation and a combination.
Consider a group of 10 people. Without doing any computation, explain why the number of committees of six people is equal to the number of committees of four people.
Without doing any computation, explain why C(10, 3) = C(10, 7).
Without doing any computation, explain why C(10, 4) + C(10, 5) = C(11, 5). Hint: Suppose that a committee of size five is to be chosen from a pool of 11 people, and John Doe is one of the people. How many committees are there that include John? How many committees are there that don't include John?
Sixty people with a certain medical condition were given pills. Fifteen of these people received placebos. Forty people showed improvement, and 30 of these had received an actual drug. How many of the people who received the drug showed no improvement?
1. For what values of n and r does Pascal's formula say that the number 10 in the triangle is the sum of the numbers 4 and 6?2. Derive Pascal's formula from the fact thatIn the following triangular table, known as Pascal's triangle, the entries in the nth row are the binomial coefficients Observe
Derive Pascal's formula from the fact that C (n, r) is the number of ways of selecting r objects from a set of n objects. Let x denote the nth object of the set. Count the number of ways that a subset of r objects containing x can be selected, and then count the number of ways that a subset of r
Use Pascal's formula to extend Pascal's triangle to the 12th row. Determine the values offrom the extended triangle. In the following triangular table, known as Pascal's triangle, the entries in the nth row are the binomial coefficients Observe that each number (other than the ones) is the sum of
(a) Show that, for any positive integer n,
(a) Show that, for any positive integer n,1 + 2 + 4 + 8 +g+ 2n = 2n+1 - 1.(b) Show that the sum of the elements of any row of Pascal's triangle equals one more than the sum of the elements of all previous rows.In the following triangular table, known as Pascal's triangle, the entries in the nth row
(a) Consider the 7th row of Pascal's triangle. Observe that each interior number (that is, a number other than 1) is divisible by 7. For what values of n, for 1 ‰¤ n ‰¤ 12, are the interior numbers of the nth row divisible by n?(b) Confirm that each of the values of n from
There are four odd numbers in the 6th row of Pascal's triangle (1, 15, 15, 1), and 4 = 22 is a power of 2. For the 0th through 12th rows of Pascal's triangle, show that the number of odd numbers in each row is a power of 2.In Fig. 1(a), each odd number in the first eight rows of Pascal's triangle
Assume that the property shown in Figs. 1 and 2 continues to hold for subsequent rows of Pascal's triangle. Use this result to explain why the number of odd numbers in each row of Pascal's triangle is a power of 2.In the following triangular table, known as Pascal's triangle, the entries in the nth
1. What is a set? 2. What is a subset of a set? 3. What is an element of a set? 4. Define a universal set. 5. Define the empty set. 6. Define the complement of the set A. 7. Define the intersection of the two sets A and B. 8. Define the union of the two sets A and B.
1. Give a formula that can be used to calculate each of the following:2. State the binomial theorem.
1. If a set contains n elements, how many subsets does it have? 2. Explain what is meant by an ordered partition of a set. 3. Explain how to calculate the number of ordered partitions of a set.
1. State the generalized multiplication principle for counting. 2. What is meant by a permutation of n items taken r at a time? 3. How would you calculate the number of permutations of n items taken r at a time? 4. What is the difference between a permutation and a combination? 5. How would you
1. A committee of two people is to be selected from five people, R, S, T, U, and V. (a) What is the sample space for this experiment? (b) Describe the event "R is on the committee" as a subset of the sample space. (c) Describe the event "neither R nor S is on the committee" as a subset of the
An experiment consists of selecting a car at random from a college parking lot and observing the color and make. Let E be the event "the car is red," F be the event "the car is a Chevrolet," G be the event "the car is a green Ford," and H be the event "the car is black or a Chrysler." (a) Which of
1. Let S = {1, 2, 3, 4, 5, 6} be a sample space, E = {1, 2} F = {2, 3} G = {, 5, 6}. (a) Are E and F mutually exclusive? (b) Are F and G mutually exclusive? 2. Draw the events E and E on two separate Venn diagrams. Are E and E mutually exclusive? 3. Let S = {a, b, c} be a sample space. Determine
1. Let S = {1, 2, 3, 4} be a sample space, E = {1}, and F = {2, 3}. Are the events E´ ( F' and E' ( F' mutually exclusive? 2. Let S be any sample space, and E, F any events associated with S. Are the events E' ( F' and E' ( F' mutually exclusive? (Apply De Morgan's laws.) 3. Suppose that 10 coins
An experiment consists of observing the eye color and age of all United States citizens. Let E be the event "blue eyes," F the event "at least 18 years old," and G the event "brown eyes and younger than 18." (a) Are E and F mutually exclusive? (b) Are E and G mutually exclusive? (c) Are F and G
Consider the experiment and events of Exercise 19. Describe the following events: Refer to Exercise 19, An experiment consists of observing the eye color and age of all United States citizens. Let E be the event "blue eyes," F the event "at least 18 years old," and G the event "brown eyes and
1. Suppose that you observe the number of passengers arriving at a metro station on a shuttle bus that holds up to eight passengers. Describe the sample space? 2. Dice A pair of dice is rolled, and the sum of the numbers on the two uppermost faces is observed. What is the sample space of this
1. In the NBA, the 14 basketball teams that did not make the playoffs participate in the draft lottery. Ping-pong balls numbered 1 through 14 are placed in a lottery machine, and a sample of four balls is drawn randomly to determine which team will have the first overall draft pick. The order in
Clue is a board game in which players are given the opportunity to solve a murder that has six suspects, six possible weapons, and nine possible rooms where the murder may have occurred. The six suspects are Colonel Mustard, Miss Scarlet, Professor Plum, Mrs. White, Mr. Green, and Mrs. Peacock.
1. An experiment consists of tossing a coin two times and observing the sequence of heads and tails.(a) What is the sample space of this experiment?(b) Describe the event "the first toss is a head" as a subset of the sample space?2. A pair of four-sided dice-each with the numbers from 1 to 4 on
1. Suppose that we have two urns-call them urn I and urn II-each containing red balls and white balls. An experiment consists of selecting an urn and then selecting a ball from that urn and noting its color. (a) What is a suitable sample space for this experiment? (b) Describe the event "urn I is
An efficiency expert records the time it takes an assembly line worker to perform a particular task. Let E be the event "more than 5 minutes," F the event "less than 8 minutes," and G the event "less than 4 minutes." (a) Describe the sample space for this experiment. (b) Describe the events E ( F,
1. A census taker records the annual income of each household they visit. Let E be the event "more than $50,000" and F be the event "less than $75,000." Describe the events E( F, E', and F'. 2. A campus survey is taken to correlate the number of years that students have been on campus with their
A box contains a penny, a nickel, a dime, a quarter, and a half dollar. You select two coins at random from the box. (a) Construct a sample space for this situation. (b) List the elements of the event E in which the total value of the coins you have selected is an even number of cents.
1. The probability that the average major league baseball player will get a hit when at bat is .25. What are the odds that the average major league baseball player will get a hit? 2. A committee consists of five men and five women. If three people are selected at random from the committee, what is
1. A drawer contains two red socks and two blue socks. If two socks are drawn randomly from the drawer, what is the probability that the two socks have the same color?
1. Five of the apples in a barrel of 100 apples are rotten. If four apples are selected from the barrel, what is the probability that at least two of the apples are rotten? 2. Of the nine city council members, four favor school vouchers and five are opposed. If a subcommittee of three council
Prior to taking an essay examination, students are given 10 questions to prepare. Six of the ten will appear on the exam. One student decides to prepare only eight of the questions. Assume that the questions are equally likely to be chosen by the professor. (a) What is the probability that they
1. A coin is to be tossed five times. What is the probability of obtaining at least one head? 2. Two players each toss a coin three times. What is the probability that they get the same number of tails?
1. In an Olympic swimming event, two of the seven contestants are American. The contestants are randomly assigned to lanes 1 through 7. What is the probability that the Americans are assigned to the first two lanes? 2. Sixty-eight men's college basketball teams compete in the NCAA championship
1. Some of the candidates for president of the computer club at Riverdale High are seniors, and the rest are juniors. Let J be the event in which a junior is elected, and let F be the event in which a female is elected. Describe the following events: a. J ∩ F' b. (J ∩ F)' c. J ∪ F' 2. Suppose
1. A card is drawn at random from a deck of cards. Then the card is replaced, and the deck is thoroughly shuffled. This process is repeated two more times. (a) What is the probability that all three cards are aces? (b) What is the probability that at least one of the cards is an ace?
What is the probability of having each of the numbers one through six appear in six consecutive rolls of a die?
Find the odds in favor of getting four different numbers when tossing four dice.
1. What is the probability that, out of a group of five people, at least two people have the same birthday? Note: Assume that there are 365 days in a year. 2. Four people are chosen at random. What is the probability that at least two of them were born on the same day of the week?
1. Let E and F be events with Pr (E) = .4, Pr(F) = .3, and Pr(E ∪ F) = .5. Find Pr(E | F). 2. Let E and F be events with Pr (E ∩ F) = 1/10 and Pr (E | F) = 1/17 Find Pr (F).
When a coin is tossed three times, what is the probability of at least one tail appearing, given that at least one head appeared?
1. Suppose that a pair of dice is rolled. Given that the two numbers are different, what is the probability that one die shows a three?2. Consider Table 1, with figures in millions, pertaining to the 2015 American civilian labor force (age 20+). Find the probability that a person selected at random
Out of the 50 colleges in a certain state, 25 are private, 15 offer engineering majors, and 5 are private colleges offering engineering majors. Find the probability that a college selected at random from the state (a) offers an engineering major. (b) offers an engineering major, given that it is
1. Suppose that a certain college contains an equal number of female and male students and that 8% of the female population are premed majors. What is the probability that a randomly selected student is a female premed major? 2. An urn contains 10 red balls and 20 green balls. If four balls are
1. A red die and a green die are rolled as a pair. Let E be the event that "the red die shows a 2" and let F be the event that "the sum of the numbers is 8." Are the events E and F independent? 2. Suppose that we toss a coin three times and observe the sequence of heads and tails. Let E be the
1. Suppose that E and F are mutually exclusive events with Pr(E) = .5 and Pr(F) = .3. Find Pr(E ∪ F). 2. Of the 120 students in a class, 30 speak Chinese, 50 speak Spanish, 75 speak French, 12 speak Spanish and Chinese, 30 speak Spanish and French, and 15 speak Chinese and French. Seven students
1. Fred will do well on his final exam if the exam is easy or if he studies hard. Suppose that the probability is .4that the exam is easy, the probability is .75 that Fred will study hard, and that the two events are independent. What is the probability that (a) the exam is easy and that Fred will
1. Let A and B be independent events with Pr (A) = .3 and Pr (B) = .4. What is the probability that exactly one of the events A or B occurs? 2. Each box of a certain brand of candy contains either a toy airplane or a toy boat. If one-third of the boxes contain an airplane and two-thirds contain a
1. An urn contains three balls numbered 1, 2, and 3. Balls are drawn one at a time without replacement until the sum of the numbers drawn is four or more. Find the probability of stopping after exactly two balls are drawn. 2. A carnival huckster has placed a coin under one of three cups and asks
1. Of a group of people surveyed in a political poll, 60% said that they would vote for candidate R. Of those who said that they would vote for R, 90% actually voted for R, and of those who did not say that they would vote for R, 5% actually voted for R. What percent of the group voted for R? 2.
An auditing procedure for income tax returns has the following characteristics: If the return is incorrect, the probability is 90% that it will be rejected; if the return is correct, the probability is 95% that it will be accepted. Suppose that 80% of all income tax returns are correct. If a return
A supermarket has three employees who package and weigh produce. Employee A records the correct weight 98% of the time. Employees B and C record the correct weight 97% and 95% of the time, respectively. Employees A, B, and C handle 40%, 40%, and 20% of the packaging, respectively. A customer
An island contains an equal number of one-headed, two-headed, and three-headed dragons. If a dragon head is picked at random, what is the likelihood of its belonging to a one-headed dragon?
1. Explain why it makes sense that if E and F are independent events, then so are E and F .
What additional information would allow you to compute Pr (E ¨ F ) if you already know Pr (E ) and Pr (F )?
Explain why two independent events with nonzero probabilities cannot be mutually exclusive.
Explain why two mutually exclusive events with nonzero probabilities cannot be independent.
1. (True or False) When a conditional probability is calculated, the probability of the event that is given is placed in the denominator. 2. Can two events E and F be mutually exclusive if Pr (E) = .6 and Pr (F) = .7?
1. The odds of an American worker living within 20 minutes of work are 13 to 12. What is the probability that a worker selected at random lives within 20 minutes of work? Lives more than 20 minutes from work? 2. According to the Professional Golfers Association (PGA), the odds of a professional
Classify the type of probability as logical, empirical, or judgmental. 1. Pr (China will win the most gold medals in the 2020 Olympics) 2. Pr (a person who smokes two packs of cigarettes each day has an increased risk of lung cancer) 3. Pr (obtaining a sum of 7 when rolling a pair of dice) 4. Pr
Of the 193 member countries of the United Nations, 54 are in the African Group and 23 are in the Eastern European Group. Suppose that a country is selected at random from the members of the United Nations. (a) What is the probability that the country is in the African Group? (b) What is the
An experiment consists of selecting a letter at random from the alphabet. Find the probability that the letter selected (a) Precedes G alphabetically. (b) Is a vowel (A, E, I, O, or U). (c) Precedes G alphabetically or is a vowel?
An experiment consists of selecting a number at random from the set of numbers {1, 2, 3, 4, 5, 6, 7, 8, 9}. Find the probability that the number selected is (a) Less than 4. (b) Odd. (c) Less than 4 or odd.
Suppose that a red die and a green die are rolled and the numbers on the sides that face upward are observed. (See Example 7 of this section and Example 2 of the first section.) (a) What is the probability that the numbers add up to 9? (b) What is the probability that the sum of the numbers is less
1. An experiment consists of observing the genders and birth orders in three-child families. Assume all eight outcomes have the same probability of occurring. (a) What is the probability that a family has at least two boys? (b) What is the probability that the oldest child is a girl? 2. The given

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