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  • Chapter 12Sets and Counting: Homework

    SETS AND COUNTING

    Find the indicated sets.

    List all subsets of the following set.

    (12.1){Al, Bob}
    (12.2){Al, Bob},{Al},{Bob}, Ø
    Exercise 2.

    List all subsets of the following set.

    (12.3){Al, Bob, Chris}

    List the elements of the following set.

    (12.4){Al, Bob, Chris, Dave}∩{Bob, Chris, Dave, Ed}
    (12.5){Bob, Chris, Dave}
    Exercise 4.

    List the elements of the following set.

    (12.6){Al, Bob, Chris, Dave}∪{Bob, Chris, Dave, Ed}

    In Problems 5 - 8, let Universal set=U={a,b,c,d,e,f,g,h,i,j}, V={a,e,i,f,h}, and W={a,c,e,g,i}.

    List the members of the following sets.

    (12.7)VW
    (12.8){a,e,i,f,h,c,g}
    Exercise 6.
    (12.9)VW
    (12.10)
    _autogen-svg2png-0013.png
    (12.11){b,d,j}
    Exercise 8.
    (12.12)
    _autogen-svg2png-0015.png

    In 9 - 12, let Universal set=U={1,2,3,4,5,6,7,8,9,10}, A={1,2,3,4,5}, B={1,3,4,6}, and C={2,4,6}.

    List the members of the following sets.

    (12.13)AB
    (12.14){1,2,3,4,5,6}
    Exercise 10.
    (12.15)AC
    (12.16)
    _autogen-svg2png-0023.png
    Ø
    Exercise 12.
    (12.17)
    _autogen-svg2png-0024.png

    Find the number of elements in the following sets.

    In Mrs. Yamamoto's class of 35 students, 12 students are taking history, 18 are taking English, and 4 are taking both. Draw a Venn diagram and determine how many students are taking neither history nor English?

    9 students
    Exercise 14.

    In the County of Santa Clara 700,000 people read the San Jose Mercury News, 400,000 people read the San Francisco Examiner, and 100,000 read both newspapers. How many read either the Mercury News or the Examiner?

    A survey of athletes revealed that for their minor aches and pains, 30 used aspirin, 50 used ibuprofen, and 15 used both. How many athletes were surveyed?

    65
    Exercise 16.

    In a survey of computer users, it was found that 50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use at least one of these Brands?

    This quarter, a survey of 100 students at De Anza college finds that 50 take math, 40 take English, and 30 take history. Of these 15 take English and math, 10 take English and history, 10 take math and history, and 5 take all three subjects. Draw a Venn diagram and determine the following.

    1. The number of students taking math but not the other two subjects.

    2. The number of students taking English or math but not history.

    3. The number of students taking none of these subjects.

    1. 30

    2. 60

    3. 10

    Exercise 18.

    In a survey of investors it was found that 100 invested in stocks, 60 in mutual funds, and 50 in bonds. Of these, 35 invested in stocks and mutual funds, 30 in mutual funds and bonds, 28 in stocks and bonds, and 20 in all three. Determine the following.

    1. The number of investors that participated in the survey.

    2. How many invested in stocks or mutual funds but not in bonds?

    3. How many invested in exactly one type of investment?

    TREE DIAGRAMS AND THE MULTIPLICATION AXIOM

    Do the following problems using a tree diagram or the multiplcation axiom.

    A man has 3 shirts, and 2 pairs of pants. Use a tree diagram to determine the number of possible outfits.

    6
    Exercise 20.

    In a city election, there are 2 candidates for mayor, and 3 for supervisor. Use a tree diagram to find the number of ways to fill the two offices.

    There are 4 roads from Town A to Town B, 2 roads from Town B to Town C. Use a tree diagram to find the number of ways one can travel from Town A to Town C.

    8
    Exercise 22.

    Brown Home Construction offers a selection of 3 floor plans, 2 roof types, and 2 exterior wall types. Use a tree diagram to determine the number of possible homes available.

    For lunch, a small restaurant offers 2 types of soups, three kinds of sandwiches, and two types of soft drinks. Use a tree diagram to determine the number of possible meals consisting of a soup, sandwich, and a soft drink.

    12
    Exercise 24.

    A California license plate consists of a number from 1 to 5, then three letters followed by three digits. How many such plates are possible?

    A license plate consists of three letters followed by three digits. How many license plates are possible if no letter may be repeated?

    15,600,000
    Exercise 26.

    How many different 4-letter radio station call letters can be made if the first letter must be K or W and none of the letters may be repeated?

    How many seven-digit telephone numbers are possible if the first two digits cannot be ones or zeros?

    6,400,000
    Exercise 28.

    How many 3-letter word sequences can be formed using the letters {a,b,c,d} if no letter is to be repeated?

    A family has two children, use a tree diagram to determine all four possibilities.

    BB, BG, GB, GG
    Exercise 30.

    A coin is tossed three times and the sequence of heads and tails is recorded. Use a tree diagram to determine the different possibilities.

    In how many ways can a 4-question true-false test be answered?

    16
    Exercise 32.

    In how many ways can three people be made to stand in a straight line?

    A combination lock is opened by first turning to the left, then to the right, and then to the left again. If there are 30 digits on the dial, how many possible combinations are there?

    27,000
    Exercise 34.

    How many different answers are possible for a multiple-choice test with 10 questions and five possible answers for each question?

    PERMUTATIONS

    Do the following problems using permutations.

    How many three-letter words can be made using the letters {a,b,c,d,e} if no repetitions are allowed?

    60
    Exercise 36.

    A grocery store has five checkout counters, and seven clerks. How many different ways can the clerks be assigned to the counters?

    A group of fifteen people who are members of an investment club wish to choose a president, and a secretary. How many different ways can this be done?

    210
    Exercise 38.

    Compute the following.

    1. 9P2

    2. 6P4

    3. 8P3

    4. 7P4

    In how many ways can the letters of the word CUPERTINO be arranged if each letter is used only once in each arrangement?

    362,880
    Exercise 40.

    How many permutations of the letters of the word PROBLEM end in a vowel?

    How many permutations of the letters of the word SECURITY end in a consonant?

    25,200
    Exercise 42.

    How many permutations of the letters PRODUCT have consonants in the second and third positions?

    How many three-digit numbers are there?

    900
    Exercise 44.

    How many three-digit odd numbers are there?

    In how many different ways can five people be seated in a row if two of them insist on sitting next to each other?

    48
    Exercise 46.

    In how many different ways can five people be seated in a row if two of them insist on not sitting next to each other?

    In how many ways can 3 English, 3 history, and 2 math books be set on a shelf, if the English books are set on the left, history books in the middle, and math books on the right?

    72
    Exercise 48.

    In how many ways can 3 English, 3 history, and 2 math books be set on a shelf, if they are grouped by subject?

    You have 5 math books and 6 history books to put on a shelf with five slots. In how many ways can you put the books on the shelf if the first two slots are to be filled with math books and the next three with history books?

    2,400
    Exercise 50.

    You have 5 math books and 6 history books to put on a shelf with five slots. In how many ways can you put the books on the shelf if the first two slots are to be filled with the books of one subject and the next three slots are to be filled with the books of the other subject?

    CIRCULAR PERMUTATIONS AND PERMUTATIONS WITH SIMILAR ELEMENTS

    Do the following problems using the techniques learned in this section.

    In how many different ways can five children hold hands to play "Ring Around the Rosy"?

    24
    Exercise 52.

    In how many ways can three people be made to sit at a round table?

    In how many different ways can six children ride a "Merry Go Around" with six horses?

    120
    Exercise 54.

    In how many ways can three couples be seated at a round table, so that men and women sit alternately?

    In how many ways can six trinkets be arranged on a chain?

    120
    Exercise 56.

    In how many ways can five keys be put on a key ring?

    Find the number of different permutations of the letters of the word MASSACHUSETTS.

    64,864,800
    Exercise 58.

    Find the number of different permutations of the letters of the word MATHEMATICS.

    Seven flags, three red, two white, and two blue, are to be flown on seven poles. How many different arrangements are possible?

    210
    Exercise 60.

    How many different ways can three pennies, two nickels and five dimes be arranged in a row?

    How many four-digit numbers can be made using two 2's and two 3's?

    6
    Exercise 62.

    How many five-digit numbers can be made using two 6's and three 7's?

    If a coin is tossed 5 times, how many different outcomes of 3 heads, and 2 tails are possible?

    10
    Exercise 64.

    If a coin is tossed 10 times, how many different outcomes of 7 heads, and 3 tails are possible?

    If a team plays ten games, how many different outcomes of 6 wins, and 4 losses are possible?

    210
    Exercise 66.

    If a team plays ten games, how many different ways can the team have a winning season?

    COMBINATIONS

    Do the following problems using combinations.

    How many different 3-people committees can be chosen from ten people?

    120
    Exercise 68.

    How many different 5-player teams can be chosen from eight players?

    In how many ways can a person choose to vote for three out of five candidates on a ballot for a school board election?

    10
    Exercise 70.

    Compute the following:

    1. 9C2

    2. 6C4

    3. 8C3

    4. 7C4

    How many 5-card hands can be chosen from a deck of cards?

    2,598,960
    Exercise 72.

    How many 13-card bridge hands can be chosen from a deck of cards?

    There are twelve people at a party. If they all shake hands, how many different hand-shakes are there?

    66
    Exercise 74.

    In how many ways can a student choose to do four questions out of five on a test?

    Five points lie on a circle. How many chords can be drawn through them?

    10
    Exercise 76.

    How many diagonals does a hexagon have?

    There are five teams in a league. How many games are played if every team plays each other twice?

    20
    Exercise 78.

    A team plays 15 games a season. In how many ways can it have 8 wins and 7 losses?

    In how many different ways can a 4-child family have 2 boys and 2 girls?

    6
    Exercise 80.

    A coin is tossed five times. In how many ways can it fall three heads and two tails?

    The shopping area of a town is a square that is six blocks by six blocks. How many different routes can a taxi driver take to go from one corner of the shopping area to the opposite cater-corner?

    924
    Exercise 82.

    If the shopping area in Exercise 81. has a rectangular form of 5 blocks by 3 blocks, then how many different routes can a taxi driver take to drive from one end of the shopping area to the opposite kitty corner end?

    COMBINATIONS INVOLVING SEVERAL SETS

    Following problems involve combinations from several different sets.

    How many 5-people committees consisting of three boys and two girls can be chosen from a group of four boys and four girls?

    24
    Exercise 84.

    A club has 4 men, 5 women, 8 boys and 10 girls as members. In how many ways can a group of 2 men, 3 women, 4 boys and 4 girls be chosen?

    How many 4-people committees chosen from four men and six women will have at least three men?

    25
    Exercise 86.

    A batch contains 10 transistors of which three are defective. If three are chosen, in how many ways can one get two defective?

    In how many ways can five counters labeled A, B, C, D and E at a store be staffed by two men and three women chosen from a group of four men and six women?

    14,400
    Exercise 88.

    How many 4-letter word sequences consisting of two vowels and two consonants can be made from the letters of the word PHOENIX if no letter is repeated?

    Three marbles are chosen from an urn that contains 5 red, 4 white, and 3 blue marbles. How many samples of the following type are possible?

    All three white.

    4
    Exercise 90.

    Two blue and one white.

    One of each color.

    60
    Exercise 92.

    All three of the same color.

    At least two red.

    80
    Exercise 94.

    None red.

    Five coins are chosen from a bag that contains 4 dimes, 5 nickels, and 6 pennies. How many samples of five of the following type are possible?

    At least four nickels.

    51
    Exercise 96.

    No pennies.

    Five of a kind.

    7
    Exercise 98.

    Four of a kind.

    Two of one kind and two of another kind.

    1,410
    Exercise 100.

    Three of one kind and two of another kind.

    Find the number of different ways to draw a 5-card hand from a deck to have the following combinations.

    Three face cards.

    171,600
    Exercise 102.

    A heart flush(all hearts).

    Two hearts and three diamonds.

    22,308
    Exercise 104.

    Two cards of one suit, and three of another suit.

    Two kings and three queens.

    24
    Exercise 106.

    Two cards of one value and three of another value.

    BINOMIAL THEOREM

    Use the Binomial Theorem to do the following problems.

    Expand (a+b)5.

    (12.18)
    _autogen-svg2png-0036.png
    Exercise 108.

    Expand (ab)6.

    Expand (x−2y)5.

    (12.19)
    _autogen-svg2png-0039.png
    Exercise 110.

    Expand (2x−3y)4.

    Find the third term of (2x−3y)6.

    (12.20)2160x4y2
    Exercise 112.

    Find the sixth term of (5x+y)8.

    Find the coefficient of the x3y4 term in the expansion of (2x+y)7.

    280
    Exercise 114.

    Find the coefficient of the a4b6 term in the expansion of (3a−b)10.

    A coin is tossed 5 times, in how many ways is it possible to get three heads and two tails?

    10
    Exercise 116.

    A coin is tossed 10 times, in how many ways is it possible to get seven heads and three tails?

    How many subsets are there of a set that has 6 elements?

    64
    Exercise 118.

    How many subsets are there of a set that has n elements?

    CHAPTER REVIEW

    Suppose of the 4,000 freshmen at a college everyone is enrolled in a mathematics or an English class during a given quarter. If 2,000 are enrolled in a mathematics class, and 3,000 in an English class, how many are enrolled in both a mathematics class and an English class?

    1,000

    In a survey of 250 people, it was found that 125 had read Time magazine, 175 had read Newsweek, 100 had read U. S. News, 75 had read Time and Newsweek, 60 had read Newsweek and U. S. News, 55 had read Time and U. S. News, and 25 had read all three.

    1. How many had read Time but not the other two?

    2. How many had read Time or Newsweek but not the U. S. News And World Report?

    3. How many had read none of these three magazines?

    1. 20

    2. 135

    3. 15

    At a manufacturing plant, a product goes through assembly, testing, and packing. If a plant has three assembly stations, two testing stations, and two packing stations, in how many different ways can a product achieve its completion?

    12

    Six people are to line up for a photograph. How many different lineups are possible if three of them insist on standing next to each other ?

    144

    How many four-letter word sequences can be made from the letters of the word CUPERTINO?

    3,024

    In how many different ways can a 20-question multiple choice test be designed so that its answers contain 2 A's, 4 B's, 9 C's, 3 D's, and 2 E's?

    11,639,628,000

    The U. S. Supreme Court has nine judges. In how many different ways can the judges cast a six-to-three decision in favor of a ruling?

    84

    In how many different ways can a coach choose a linebacker, a guard, and a tackle from five players on the bench, if all five can play any of the three positions?

    60

    How many three digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if no repetitions are allowed?

    24

    Compute:

    1. 9C4

    2. 8P3

    3. _autogen-svg2png-0051.png

    1. 126

    2. 336

    3. 210

    In how many ways can 3 English, 3 Math, and 4 Spanish books be set on a shelf if the books are grouped by subject?

    5,184

    In how many ways can a 10-question multiple choice test with four possible answers for each question be answered?

    1,048,576

    On a soccer team three fullbacks can play any of the three fullback positions, left, center, and right. The three halfbacks can play any of the three halfback positions, the four forwards can play any of the four positions, and the goalkeeper plays only his position. How many different arrangements of the 11 players are possible?

    46,200

    From a group of 6 people, 3 are assigned to cleaning, 2 to hauling and one to garbage collecting. How many different ways can this be done?

    60

    How many three-letter word sequences can be made from the letters of the word OXYGEN?

    120

    In how many ways can 3 books be selected from 4 English and 2 History books if at least one English book must be chosen?

    20

    Five points lie on the rim of a circle. Choosing the points as vertices, how many different triangles can be drawn?

    10

    A club consists of six men and nine women. In how many ways can a president, a vice president and a treasurer be chosen if the two of the officers must be women?

    1296

    Of its 12 sales people, a company wants to assign 4 to its Western territory, 5 to its Northern territory, and 3 to its Southern territory. How many ways can this be done?

    27,720

    How many permutations of the letters of the word OUTSIDE have consonants in the first and last place?

    720

    How many distinguishable permutations are there in the word COMMUNICATION?

    194,594,400

    How many five-card poker hands consisting of the following distribution are there?

    1. A flush(all five cards of a single suit)

    2. Three of a kind(e.g. three aces and two other cards)

    3. Two pairs(e.g. two aces, two kings and one other card)

    4. A straight(all five cards in a sequence)

    1. 5148

    2. 58,656

    3. 123,552

    4. 10,240 or 9216

    Company stocks on an exchange are given symbols consisting of three letters. How many different three-letter symbols are possible?

    17,576

    How many four-digit odd numbers are there?

    4500

    In how many ways can 7 people be made to stand in a straight line? In a circle?

    5040; 720

    A united nations delegation consists of 6 Americans, 5 Russians, and 4 Chinese. Answer the following questions.

    1. How many committees of five people are there?

    2. How many committees of three people consisting of at least one American are there?

    3. How many committees of four people having no Russians are there?

    4. How many committees of three people have more Americans than Russians?

    5. How many committees of three people do not have all three Americans?

    1. 3003

    2. 371

    3. 210

    4. 191

    5. 435

    If a coin is flipped five times, in how many different ways can it show up three heads?

    10

    To reach his destination, a man is to walk three blocks north and four blocks west. How many different routes are possible?

    35

    All three players of the women's beach volleyball team, and all three players of the men's beach volleyball team are to line up for a picture with all members of the women's team lined together and all members of men's team lined up together. How many ways can this be done?

    72

    From a group of 6 Americans, 5 Japanese and 4 German delegates, two Americans, two Japanese and a German are chosen to line up for a photograph. In how many different ways can this be done?

    72,000

    Find the fourth term of the expansion (2x−3y)8.

    (12.21)−48384x5y3

    Find the coefficient of the a5b4 term in the expansion of (a−2b)9.

    (12.22)2016a5b4
    Solutions