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

Use a Venn diagram to find the number of people in the sets. 1. L ∩ (F ∪ S) 2. (L ∪ F ∪ S) 3. L' 4. L ∪ S ∪ F' 5. F ∩ S' ∩ L' A survey in a local high school shows that, of the 4000 students in the school, 2000 take French (F) 3000 take Spanish (S) 500 take Latin (L) 1500 take
One hundred college students were surveyed after voting in an election involving a Democrat and a Republican. There were 50 first-year students, 55 voted Democratic, and 25 were non-first-year students who voted Republican. How many first-year students voted Democratic?
A group of 100 workers were asked whether they were college graduates and whether they belonged to a union. According to their responses, 60 were not college graduates, 20 were nonunion college graduates, and 30 were union members. How many of the workers were neither college graduates nor union
A class of 30 students was given a diagnostic test on the first day of a mathematics course. At the end of the semester, only 2 of the 21 students who had passed the diagnostic test failed the course. A total of 23 students passed the course. How many students managed to pass the course even though
A group of applicants for training as air-traffic controllers consists of 35 pilots, 20 veterans, 30 pilots who were not veterans, and 50 people who were neither veterans nor pilots. How large was the group?
How many of the students are either seniors or biology majors? A group of 61 students has the following characteristics: 6 are biology majors and seniors 17 are biology majors and not seniors 12 are not seniors and are majoring in a field other than biology.
How many of the students are seniors? A group of 61 students has the following characteristics: 6 are biology majors and seniors 17 are biology majors and not seniors 12 are not seniors and are majoring in a field other than biology.
How many of the students are not seniors? A group of 61 students has the following characteristics: 6 are biology majors and seniors 17 are biology majors and not seniors 12 are not seniors and are majoring in a field other than biology.
How many of the students are biology majors? A group of 61 students has the following characteristics: 6 are biology majors and seniors 17 are biology majors and not seniors 12 are not seniors and are majoring in a field other than biology.
How many of the seniors are not biology majors? A group of 61 students has the following characteristics: 6 are biology majors and seniors 17 are biology majors and not seniors 12 are not seniors and are majoring in a field other than biology.
How many of the students are not biology majors? A group of 61 students has the following characteristics: 6 are biology majors and seniors 17 are biology majors and not seniors 12 are not seniors and are majoring in a field other than biology.
How many students like rock only? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not country 9 like rock and
How many students like country but not rock? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not country 9 like
How many students like rap and country but not rock? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not
How many students like rap or country but not rock? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not
How many students like exactly one of the genres? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not country 9
How many students like all three genres? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not country 9 like rock
How many students like at least two of the three genres? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not
How many students do not like either rock or country? A campus radio station surveyed 190 students to determine the genres of music they liked. The survey results follow: 114 like rock 50 like country 15 like rock and rap 11 like rap and country 20 like rap only 10 like rock and rap, but not
One hundred and sixty business executives were surveyed to determine whether they regularly visit the CNN Money, Bloomberg, or The Wall Street Journal websites. The survey showed that 70 visit CNN Money, 60 visit Bloomberg, 55 visit The Wall Street Journal, 45 visit exactly two of the three
A survey of the characteristics of 100 small businesses that had failed revealed that 95 of them either were undercapitalized, had inexperienced management, or had a poor location. Four of the businesses had all three of these characteristics. Forty businesses were undercapitalized but had
Each of the 100 students attending a conservatory of music plays at least one of three instruments: piano, violin, and clarinet. Of the students, 65 play the piano, 42 play the violin, 28 play the clarinet, 20 play the piano and the violin, 10 play the violin and the clarinet, and 8 play the piano
Students living in a certain dormitory were asked about their enrollment in mathematics and history courses. Ten percent were taking both types of courses, and twenty percent were taking neither type of course. One hundred sixty students were taking a mathematics course but not a history course,
1. Jolene wants to drive from her house to the grocery store and then to the library. If her GPS suggests four routes from her house to the grocery store, and two routes from the grocery store to the library, how many total ways are there for Jolene to do this?2. There are three bridges from the
1. A group of five boys and three girls is to be photographed. (a) How many ways can they be arranged in one row? (b) How many ways can they be arranged with the girls in the front row and the boys in the back row? 2. Three history books and six novels are to be arranged on a bookshelf. (a) How
1. How many different three-letter words (including nonsense words) are there in which successive letters are different? 2. How many different outfits consisting of a coat and a hat can be selected from two coats and three hats? 3. How many different outfits can be selected from two coats, four
1. How many Social Security numbers are available if the only restriction is that the number 000-00-0000 cannot be assigned? 2. In 1923, the Federal Communications Commission directed that all new radio stations east of the Mississippi River have call letters beginning with the letter W. How many
1. How many 5-digit numbers are palindromes? 2. How many 6-digit numbers are palindromes? 3. How many 4-letter words (including nonsense words) are palindromes? 4. How many 3-letter words (including nonsense words) are palindromes? A number or word is said to be a palindrome if it reads the same
1. The World Series of Baseball is played between the American League and National League champions, in which each league consists of 15 teams. How many different possible matchups are there for the World Series? 2. The Super Bowl is a game played between the National Football Conference and
1. A college of 20,000 students provides each student with an Internet account. Explain why letting each student have his or her initials as the username cannot possibly work. Assume that each person has a first name, middle name, and last name and therefore that each person's initials consist of
1. The final score in a soccer game is 6 to 4. How many different halftime scores are possible? 2. Each day, Gloria dresses in a blouse, a skirt, and shoes. She wants to wear a different combination on every day of the year. If she has the same number of blouses, skirts, and pairs of shoes, how
1. A man has five different pairs of gloves. In how many ways can he select a right-hand glove and a left-hand glove that do not match? 2. Fred has 11 different pairs of shoes. In how many ways can he put on a pair of shoes that do not match? 3. Toss a coin six times, and observe the sequence of
1. Each of the 10 questions on a multiple-choice exam has four possible answers. How many different ways are there for a student to answer the questions? Assume that every question must be answered. 2. Rework Exercise 41 under the assumption that not every question must be answered. 3. 43. How many
How many ways can eight people stand in a line for a group picture? If you took a picture every 15 seconds (day and night with no breaks), how long would it take to photograph every possible arrangement?
1. A company is manufacturing license plates with the pattern LL#-##LL, where L represents a letter and # represents a digit from 1 through 9. If a letter can be any letter from A to Z except O, how many different license plates are possible? If the company produces 500,000 license plates per week,
1. A restaurant menu lists0 7 appetizers, 10 entrées, and 4 desserts. How many ways can a diner select a three-course meal? 2. The gift-wrap desk at a large department store offers 5 box sizes, 10 wrapping papers, 7 colors of ribbon in 2 widths, and 9 special items to be added on the box.
1. How many different ways can a Venn diagram with three circles be shaded? 2. How many outcomes are possible if the first number is green? 3. How many outcomes are possible if all three numbers are red and no number repeats? An American roulette wheel consists of 38 numbered pockets. Two of them
The manager of a Little League baseball team has picked the nine starting players for a game. How many different batting orders are possible under each of the following conditions? (a) There are no restrictions. (b) The pitcher must bat last. (c) The pitcher must bat last, the catcher eighth, and
1. A physiologist wants to test the effects of exercise and meditation on blood pressure. She devises four different exercise programs and three different meditation programs. If she wants 10 subjects for each combination of exercise and meditation program, how many volunteers must she recruit? 2.
1. Twenty horses competed in the 2016 Kentucky Derby. Assuming no ties, how many possibilities were there for the first, second, and third place finishers? 2. Twenty athletes enter an Olympic event. Assuming no ties, how many different possibilities are there for winning the Gold Medal, Silver
1. Seven candidates for mayor, four candidates for city council president, and six propositions are being put before the electorate. How many different ballots could be cast, assuming that every voter votes on each of the items? If voters can choose to leave any item blank, how many different
Calculate the values. 1. P(4, 2) 2. P(5, 1) 3. P(6, 3) 4. P(5, 4)
1. The number of different stock abbreviations for which each abbreviation consists of four letters, none repeated. 2. The number of different airport codes in which each code consists of three letters, none repeated. 3. The selection of three different flavors of ice cream (out of 29 flavors) for
1. In how many ways can four people line up in a row for a group picture? 2. In how many ways can six people line up at a single counter to order food at a fast-food restaurant? 3. How many different selections of seven books can be made from nine books? 4. A pizzeria offers five toppings for the
1. A high school student decides to apply to four of the eight lvy League colleges. In how many possible ways can the four colleges be selected? 2. In how many different ways can a committee of 5 senators be selected from the 100 members of the U.S. Senate? 3. A sportswriter makes a preseason guess
1. Suppose that you have 36 CDs and your CD player has five slots numbered 1 through 5. How many ways can you fill your CD player? 2. How many ways can you choose five out of 10 friends to invite to a dinner party? 3. A student is required to work exactly four problems from an eight-problem exam.
In a batch of 100 DVDs, seven are defective. A sample of three DVDs is to be selected from the batch. How many samples are possible? How many of the samples consist of all defective DVDs?
1. There are 17 candidates for an elected office. If 10 candidates are selected to participate in a debate, determine the total number of possible debate groups. 2. Race Winners Theoretically, assuming no ties, how many possibilities are there for first, second, and third places in a marathon race
1. How many different poker hands are there? 2. How many different poker hands consist entirely of aces and kings? 3. How many different poker hands consist entirely of clubs? A poker hand consists of 5 cards selected from a standard deck of 52 cards.
1. How many different poker hands consist entirely of red cards? A poker hand consists of 5 cards selected from a standard deck of 52 cards. 2. Five students order different sandwiches at a campus eatery. The waiter forgets who ordered what and gives out the sandwiches at random. In how many
Suppose that you own 10 sweaters. (a) How many ways can you select four of them to take on a trip? (b) How many ways can you select six of the sweaters to leave at home? (c) Explain why the answers to parts (a) and (b) are the same.
Fred has 12 different books. (a) Suppose that Fred first gives three books to Jill and then gives four of the remaining books to Jack. How many different outcomes are possible? (b) Suppose that Fred first gives four books to Jack and then gives three of the remaining books to Jill. How many
1. Conference Games In an eight-team football conference, each team plays every other team exactly once. How many games must be played? 2. League Games In a six-team softball league, each team plays every other team three times during the season. How many games must be scheduled?
1. Powerball In the Powerball lottery, five white balls are drawn out of a drum with 69 numbered white balls and then one red ball is drawn out of a drum with 26 numbered red balls. The jackpot is won by guessing all five white balls in any order and the red Powerball. Determine the number of
1. The number of possible combinations of six numbers selected from 1 to 59 is approximately _____ times the number of combinations selected from 1 through 49. (a) 2 (b) 3 (c) 10 (d) 100 2. Drawings for Lotto are held twice per week. Suppose that you decide to purchase 110 tickets for each drawing
Two children, Moe and Joe, are allowed to select candy from a plate of nine pieces of candy. Moe, being younger, is allowed to choose first but can take only two candies. Joe is then allowed to take four of the remaining candies. Joe complains that he has fewer options than Moe. Is Joe correct? How
The 12 members of the Gotham City Council consists of four members from each of the city's three wards. In how many ways can a committee of six council members be selected if the committee must contain at least one council member from each ward?
1. The student council at Gotham College is made up of four freshmen, five sophomores, six juniors, and seven seniors. A yearbook photographer would like to line up three council members from each class for a picture. How many different pictures are possible if each group of classmates stands
1. At a party, everyone shakes hands with everyone else. If 45 handshakes take place, how many people are at the party? 2. In a football league, each team plays one game against each other team in the league. If 55 games are played, how many teams are in the league? 3. A restaurant offers its
1. Specials An ice cream parlor offers a special consisting of three scoops of ice cream chosen from 16 different flavors. Duplication of flavors is allowed. For instance, one possibility is two scoops of chocolate and one scoop of vanilla. Show that there are 816 different possible options for the
(a) Calculate the number of possible lottery tickets if the player must choose five distinct numbers from 0 to 44, inclusive, where the order does not matter. The winner must match all five. (b) Calculate the number of lottery tickets if the player must choose four distinct numbers from 0 to 99,
A bridge hand contains 13 cards. (a) What percent of bridge hands contain all four aces? (b) What percent of bridge hands contain the two red kings, the two red queens, and no other kings or queens? (c) Which is more likely-a bridge hand with four aces or one with the two red kings, the two red
1. Are there more ways to order a deck of 52 cards than there are atoms on Earth? 2. Are there more ways to rearrange the 26 letters of the alphabet than there are atoms on Earth?
1. An experiment consists of tossing a coin eight times and observing the sequence of heads and tails. (a) How many different outcomes are possible? (b) How many different outcomes have exactly four heads? 2. An experiment consists of tossing a coin nine times and observing the sequence of heads
1. Refer to the map in Fig. 3. How many shortest routes are there from A to B that pass through the point C?2. Refer to the map in Fig. 2. How many shortest routes are there from A to B that pass through the point C? 3. Routes through City Streets Refer to the map in Fig. 2. How many shortest
Refer to the map in Fig. 3. The number of shortest routes from A to B is C(9, 4).(a) Observe that the number of shortest routes from A to D is C(8, 3).(b) Observe that the number of shortest routes from A to E is C(8, 4).(c) By looking at Fig. 3, explain why C (8, 3) + C (8, 4) should equal C(9,
1. A coin is tossed 10 times, and the sequence of heads and tails is observed. How many of the possible outcomes contain three heads, with no two heads adjacent to each other? 2. Four mathematics books and seven history books are arranged on a bookshelf. In how many of the possible arrangements are
An urn contains 12 numbered balls, of which 7 are red and 5 are white. A sample of 5 balls is to be selected. (a) How many different samples are possible? (b) How many samples contain all red balls? (c) How many samples contain two red balls and three white balls? (d) How many samples contain at
An urn contains 15 numbered balls, of which 6 are red and 9 are white. A sample of six balls is to be selected. (a) How many different samples are possible? (b) How many samples contain all white balls? (c) How many samples contain two red balls and four white balls? (d) How many samples contain at
A package contains 100 LED light bulbs, of which 10 are defective. A sample of five bulbs is selected at random. (a) How many different samples are there? (b) How many of the samples contain two defective bulbs? (c) How many of the samples contain at least one defective bulb?
1. A committee has four male and six female members. In how many ways can a subcommittee consisting of two males and two females be selected? 2. In how many ways can an investor put together a portfolio of five stocks and six bonds selected from her favorite nine stocks and seven bonds? 3. How many
An experiment consists of tossing a coin seven times and observing the sequence of heads and tails. (a) How many different outcomes have at least five heads? (b) How many different outcomes have at most four heads?
1. How many poker hands consist of three cards of one rank and two cards of another rank? (Such a poker hand is called a "full house.") A poker hand consists of 5 cards selected from a standard deck of 52 cards. 2. How many poker hands consist of two cards of one rank, two cards of another
1. How many 10-letter words with no repeated letters contain the five vowels in alphabetical order? 2. How many eight-letter words with no repeated letters contain all five vowels? A "word" is interpreted to be a sequence of letters. 3. In 2015, there were three women and six men on the United
1. In how many five-digit numbers (without zeros) are the digits strictly increasing when read from left to right? 2. Order In how many four-letter words (including nonsense words) using four different letters from A through J are the letters in alphabetical order? 3. Suppose that license plates
An experiment consists of tossing a coin six times and observing the sequence of heads and tails. (a) How many different outcomes have at most three heads? (b) How many different outcomes have four or more heads?
1. In how many ways can five books out of eight be selected and lined up on a bookshelf so that their page counts increase from left to right? 2. What percent of the possible outcomes resulting from tossing a coin 100 times contain exactly 50 heads? 3. What percent of the possible outcomes
In the World Series, the American League team ("A") and the National League team ("N") play until one team wins four games. Each sequence of winners can be designated by a sequence of As and Ns. For instance, NAAAA means the National League won the first game and lost the next four games. In how
1. Refer to Exercise 5. How many different sequences are possible?In the World Series, the American League team ("A") and the National League team ("N") play until one team wins four games. Each sequence of winners can be designated by a sequence of As and Ns. For instance, NAAAA means the National
1. How many bytes have exactly five ones?2. In how many of the bytes with exactly five ones are no two zeroes next to each other?A computer byte is a string of eight digits, where each digit is either a zero or a one. Two examples are 01001001 and 11001101.3. Refer to the map in Fig. 2. How many
Calculate the value1. C(18, 16)2. C(25, 24)3.4. 5.
1. How many terms are there in the binomial expansion of (x + y)19? 2. How many terms are there in the binomial expansion of (x + y)25? 3. Determine the first three terms in the binomial expansion of (x + y)10.
1. Determine the first three terms in the binomial expansion of (x + y)20.2. Determine the last three terms in the binomial expansion of (x + y)15.3. Determine the last three terms in the binomial expansion of (x + y)12.
1. Determine the middle term in the binomial expansion of (x + y)20. 2. Determine the middle term in the binomial expansion of (x + y)10. 3. Determine the coefficient of x2 in the expansion of (1 + x)4. 4. Determine the coefficient of x3 in the expansion of (2 + x)6. 5. Determine the coefficient of
1. Determine the first three terms in the binomial expansion of (x + 2y)9. 2. Determine the last three terms in the binomial expansion of (x - y)8.
1. Determine the middle term in the binomial expansion of (x - 3y)12. 2. Determine the coefficient of x4y2 in the binomial expansion of (x + 3y)6. 3. Determine the term containing y3 in the expansion of (x - 3y)7.
1. Determine the term containing y5 in the expansion of (x - 3y)8. 2. How many different subsets can be chosen from a set of eight elements? 3. How many different subsets can be chosen from a set of nine elements? 4. How many different tips could you leave in a tip jar if you had a nickel, a dime,
1. A cable TV franchise offers 20 basic channels plus a selection (at an extra cost per channel) from a collection of 5 premium channels. How many different options are available to the subscriber? 2. A salad bar offers a base of lettuce to which tomatoes, chickpeas, beets, pinto beans, olives, and
1. In how many ways can a selection of at least one tie be made from a set of eight ties? 2. In how many ways can a selection of at most five desserts be made from a dessert trolley containing six desserts? 3. Armand's Chicago Pizzeria offers thin-crust and deep-dish pizzas in 9-, 12-, and 14-inch
1. In how many ways can a selection of at most five appetizers be made from a menu containing seven appetizers? 2. In how many ways can a selection of at least two CDs be made from a set of seven CDs? 3. Students in a physics class are required to complete at least two out of a collection of eight
1. James has nine jazz CDs and ten top 40/pop CDs. How many possible combinations of CDs can he select if he decides to take at least two CDs of each type to a party? 2. Sarah has five nonfiction and six fiction books on her reading list. When packing for her summer vacation, she decides to pack at
1. Can (85) x3y4 be a term of a binomial expansion? 2. Can (85) x3 be a term of binomial expansion? 3. Show that half of the subsets of a set of five elements have an odd number of elements.
(a) Use equation (3) to show that(b) For what values of n can equation (3) be used to show that (c) Use the binomial theorem to prove the result in part (a). (d) For what values of n can the binomial theorem be used to prove the result in part (b)?
Let S be a set of n elements. Determine the number of ordered partitions of the types. 1. n = 5; (3, 1, 1) 2. n = 5; (2, 1, 2) 3. n = 6; (2, 1, 2, 1) 4. n = 6; (3, 3) 5. n = 7; (3, 2, 2) 6. n = 7; (4, 1, 2) 7. n = 12; (4, 4, 4) 8. n = 8; (3, 3, 2) 9. n = 12; (5, 3, 2, 2) 10. n = 8; (2, 2, 2, 2)
Let S be a set of n elements. Determine the number of unordered partitions of the types. 1. n = 15; (3, 3, 3, 3, 3) 2. n = 10; (5, 5) 3. n = 18; (6, 6, 6) 4. n = 12; (4, 4, 4)
1. A brokerage house regularly reports the behavior of a group of 20 stocks, each stock being reported as "up," "down," or "unchanged." How many different reports can show seven stocks up, five stocks down, and eight stocks unchanged? 2. An investment advisory service rates investments as A, AA,
1. A nautical signal consists of six flags arranged vertically on a flagpole. If a sailor has three red flags, two blue flags, and one white flag, how many different signals are possible? 2. A psychology experiment observes groups of four individuals. In how many ways can an experimenter choose 5
1. Of the nine contestants in a contest, three will receive cars, three will receive TV sets, and three will receive radios. In how many different ways can the prizes be awarded? 2. A corporation has four employees that it wants to place in high executive positions. One will become president, one
A sales representative must travel to three cities, twice each, in the next 10 days. Her nontravel days are spent in the office. In how many different ways can she schedule her travel?
1. Ten students in a physical education class are to be divided into five-member teams for a basketball game. In how many ways can the two teams be selected? 2. In how many ways can 12 sweaters be stored into three boxes of different sizes if 6 sweaters are to be stored in the large box, 4 in the

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