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Sets and Counting: Homework

Module by: UniqU, LLC. E-mail the author

Based on: Applied Finite Mathematics: Chapter 06 by Rupinder Sekhon

Summary: This chapter covers principles of sets and counting. After completing this chapter students should be able to: use set theory and venn diagrams to solve counting problems; use the multiplication axiom to solve counting problems; use permutations to solve counting problems; use combinations to solve counting problems; and use the binomial theorem to expand x+y^n.

SETS AND COUNTING

Find the indicated sets.

Exercise 1

List all subsets of the following set.

Al, BobAl, Bob size 12{ left lbrace "Al, Bob" right rbrace } {}
(1)

Solution

Al, Bob,Al,Bob, Ø Al, Bob size 12{ left lbrace "Al, Bob" right rbrace } {},Al size 12{ left lbrace "Al" right rbrace } {},Bob, Ø size 12{ left lbrace "Bob" right rbrace } {}
(2)

Exercise 2

List all subsets of the following set.

Al, Bob, ChrisAl, Bob, Chris size 12{ left lbrace "Al, Bob, Chris" right rbrace } {}
(3)

Exercise 3

List the elements of the following set.

Al, Bob, Chris, DaveBob, Chris, Dave, EdAl, Bob, Chris, DaveBob, Chris, Dave, Ed size 12{ left lbrace "Al, Bob, Chris, Dave" right rbrace intersection left lbrace "Bob, Chris, Dave, Ed" right rbrace } {}
(4)

Solution

Bob, Chris, DaveBob, Chris, Dave size 12{ left lbrace "Bob, Chris, Dave" right rbrace } {}
(5)

Exercise 4

List the elements of the following set.

Al, Bob, Chris, DaveBob, Chris, Dave, EdAl, Bob, Chris, DaveBob, Chris, Dave, Ed size 12{ left lbrace "Al, Bob, Chris, Dave" right rbrace union left lbrace "Bob, Chris, Dave, Ed" right rbrace } {}
(6)

In Problems 5 - 8, let Universal set=U=a,b,c,d,e,f,g,h,i,jUniversal set=U=a,b,c,d,e,f,g,h,i,j size 12{"Universal set"=U= left lbrace a,b,c,d,e,f,g,h,i,j right rbrace } {}, V=a,e,i,f,hV=a,e,i,f,h size 12{V= left lbrace a,e,i,f,h right rbrace } {}, and W=a,c,e,g,iW=a,c,e,g,i size 12{W= left lbrace a,c,e,g,i right rbrace } {}.

List the members of the following sets.

Exercise 5

VWVW size 12{V union W} {}
(7)

Solution

a,e,i,f,h,c,ga,e,i,f,h,c,g size 12{ left lbrace a,e,i,f,h,c,g right rbrace } {}
(8)

Exercise 6

VWVW size 12{V intersection W} {}
(9)

Exercise 7

VW¯VW¯ size 12{ {overline {V union W}} } {}
(10)

Solution

b,d,jb,d,j size 12{ left lbrace b,d,j right rbrace } {}
(11)

Exercise 8

VˉWˉVˉWˉ size 12{ { bar {V}} intersection { bar {W}}} {}
(12)

In 9 - 12, let Universal set=U=1,2,3,4,5,6,7,8,9,10Universal set=U=1,2,3,4,5,6,7,8,9,10 size 12{"Universal set"=U= left lbrace a,b,c,d,e,f,g,h,i,j right rbrace } {}, A=1,2,3,4,5A=1,2,3,4,5 size 12{A= left lbrace 1,2,3,4,5 right rbrace } {}, B=1,3,4,6B=1,3,4,6 size 12{B= left lbrace 1,3,4,6 right rbrace } {}, and C=2,4,6C=2,4,6 size 12{C= left lbrace 2,4,6 right rbrace } {}.

List the members of the following sets.

Exercise 9

ABAB size 12{A union B} {}
(13)

Solution

1,2,3,4,5,61,2,3,4,5,6 size 12{ left lbrace 1,2,3,4,5,6 right rbrace } {}
(14)

Exercise 10

ACAC size 12{A intersection C} {}
(15)

Exercise 11

AB¯CAB¯C size 12{ {overline {A union B}} intersection C} {}
(16)

Solution

Ø

Exercise 12

AˉBC¯AˉBC¯ size 12{ { bar {A}} union {overline {B intersection C}} } {}
(17)

Find the number of elements in the following sets.

Exercise 13

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?

Solution

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?

Exercise 15

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?

Solution

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?

Exercise 17

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.

Solution

  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.

Exercise 19

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

Solution

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.

Exercise 21

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.

Solution

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.

Exercise 23

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.

Solution

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?

Exercise 25

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

Solution

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?

Exercise 27

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

Solution

6,400,000

Exercise 28

How many 3-letter word sequences can be formed using the letters a,b,c,da,b,c,d size 12{ left lbrace a,b,c,d right rbrace } {} if no letter is to be repeated?

Exercise 29

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

Solution

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.

Exercise 31

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

Solution

16

Exercise 32

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

Exercise 33

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?

Solution

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.

Exercise 35

How many three-letter words can be made using the letters a,b,c,d,ea,b,c,d,e size 12{ left lbrace a,b,c,d,e right rbrace } {} if no repetitions are allowed?

Solution

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?

Exercise 37

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?

Solution

210

Exercise 38

Compute the following.

  1. 9P29P2 size 12{9P2} {}

  2. 6P46P4 size 12{6P4} {}

  3. 8P38P3 size 12{8P3} {}

  4. 7P47P4 size 12{7P4} {}

Exercise 39

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

Solution

362,880

Exercise 40

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

Exercise 41

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

Solution

25,200

Exercise 42

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

Exercise 43

How many three-digit numbers are there?

Solution

900

Exercise 44

How many three-digit odd numbers are there?

Exercise 45

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

Solution

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?

Exercise 47

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?

Solution

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?

Exercise 49

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?

Solution

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.

Exercise 51

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

Solution

24

Exercise 52

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

Exercise 53

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

Solution

120

Exercise 54

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

Exercise 55

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

Solution

120

Exercise 56

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

Exercise 57

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

Solution

64,864,800

Exercise 58

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

Exercise 59

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

Solution

210

Exercise 60

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

Exercise 61

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

Solution

6

Exercise 62

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

Exercise 63

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

Solution

10

Exercise 64

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

Exercise 65

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

Solution

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.

Exercise 67

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

Solution

120

Exercise 68

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

Exercise 69

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

Solution

10

Exercise 70

Compute the following:

  1. 9C29C2 size 12{9C2} {}

  2. 6C46C4 size 12{6C4} {}

  3. 8C38C3 size 12{8C3} {}

  4. 7C47C4 size 12{7C4} {}

Exercise 71

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

Solution

2,598,960

Exercise 72

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

Exercise 73

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

Solution

66

Exercise 74

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

Exercise 75

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

Solution

10

Exercise 76

How many diagonals does a hexagon have?

Exercise 77

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

Solution

20

Exercise 78

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

Exercise 79

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

Solution

6

Exercise 80

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

Exercise 81

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?

Solution

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.

Exercise 83

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

Solution

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?

Exercise 85

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

Solution

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?

Exercise 87

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?

Solution

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?

Exercise 89

All three white.

Solution

4

Exercise 90

Two blue and one white.

Exercise 91

One of each color.

Solution

60

Exercise 92

All three of the same color.

Exercise 93

At least two red.

Solution

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?

Exercise 95

At least four nickels.

Solution

51

Exercise 96

No pennies.

Exercise 97

Five of a kind.

Solution

7

Exercise 98

Four of a kind.

Exercise 99

Two of one kind and two of another kind.

Solution

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.

Exercise 101

Three face cards.

Solution

171,600

Exercise 102

A heart flush(all hearts).

Exercise 103

Two hearts and three diamonds.

Solution

22,308

Exercise 104

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

Exercise 105

Two kings and three queens.

Solution

24

Exercise 106

Two cards of one value and three of another value.

BINOMIAL THEOREM

Use the Binomial Theorem to do the following problems.

Exercise 107

Expand a+b5a+b5 size 12{ left (a+b right ) rSup { size 8{5} } } {}.

Solution

a5+5a4b+10a3b2+10a2b3+5ab4+b5a5+5a4b+10a3b2+10a2b3+5ab4+b5 size 12{a rSup { size 8{5} } +5a rSup { size 8{4} } b+"10"a rSup { size 8{3} } b rSup { size 8{2} } +"10"a rSup { size 8{2} } b rSup { size 8{3} } +5 ital "ab" rSup { size 8{4} } +b rSup { size 8{5} } } {}
(18)

Exercise 108

Expand ab6ab6 size 12{ left (a - b right ) rSup { size 8{6} } } {}.

Exercise 109

Expand x2y5x2y5 size 12{ left (x - 2y right ) rSup { size 8{5} } } {}.

Solution

x510x4y+40x3y280x2y3+80xy432y5x510x4y+40x3y280x2y3+80xy432y5 size 12{x rSup { size 8{5} } - "10"x rSup { size 8{4} } y+"40"x rSup { size 8{3} } y rSup { size 8{2} } - "80"x rSup { size 8{2} } y rSup { size 8{3} } +"80" ital "xy" rSup { size 8{4} } - "32"y rSup { size 8{5} } } {}
(19)

Exercise 110

Expand 2x3y42x3y4 size 12{ left (2x - 3y right ) rSup { size 8{4} } } {}.

Exercise 111

Find the third term of 2x3y62x3y6 size 12{ left (2x - 3y right ) rSup { size 8{6} } } {}.

Solution

2160x4y22160x4y2 size 12{"2160"x rSup { size 8{4} } y rSup { size 8{2} } } {}
(20)

Exercise 112

Find the sixth term of 5x+y85x+y8 size 12{ left (5x+y right ) rSup { size 8{8} } } {}.

Exercise 113

Find the coefficient of the x3y4x3y4 size 12{x rSup { size 8{3} } y rSup { size 8{4} } } {} term in the expansion of 2x+y72x+y7 size 12{ left (2x+y right ) rSup { size 8{7} } } {}.

Solution

280

Exercise 114

Find the coefficient of the a4b6a4b6 size 12{a rSup { size 8{4} } b rSup { size 8{6} } } {} term in the expansion of 3ab103ab10 size 12{ left (3a - b right ) rSup { size 8{"10"} } } {}.

Exercise 115

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

Solution

10

Exercise 116

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

Exercise 117

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

Solution

64

Exercise 118

How many subsets are there of a set that has nn size 12{n} {} elements?

CHAPTER REVIEW

Exercise 119

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?

Solution

1,000

Exercise 120

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?

Solution

  1. 20
  2. 135
  3. 15

Exercise 121

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?

Solution

12

Exercise 122

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 ?

Solution

144

Exercise 123

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

Solution

3,024

Exercise 124

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?

Solution

11,639,628,000

Exercise 125

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?

Solution

84

Exercise 126

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?

Solution

60

Exercise 127

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

Solution

24

Exercise 128

Compute:

  1. 9C49C4 size 12{9C4} {}
  2. 8P38P3 size 12{8P3} {}
  3. 10! 4!104! 10! 4!104! size 12{"10"!4! left ("10" - 4 right )!} {}

Solution

  1. 126
  2. 336
  3. 210

Exercise 129

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?

Solution

5,184

Exercise 130

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

Solution

1,048,576

Exercise 131

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?

Solution

46,200

Exercise 132

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?

Solution

60

Exercise 133

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

Solution

120

Exercise 134

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?

Solution

20

Exercise 135

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

Solution

10

Exercise 136

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?

Solution

1296

Exercise 137

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?

Solution

27,720

Exercise 138

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

Solution

720

Exercise 139

How many distinguishable permutations are there in the word COMMUNICATION?

Solution

194,594,400

Exercise 140

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)

Solution

  1. 5148
  2. 58,656
  3. 123,552
  4. 10,240 or 9216

Exercise 141

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

Solution

17,576

Exercise 142

How many four-digit odd numbers are there?

Solution

4500

Exercise 143

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

Solution

5040; 720

Exercise 144

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?

Solution

  1. 3003
  2. 371
  3. 210
  4. 191
  5. 435

Exercise 145

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

Solution

10

Exercise 146

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

Solution

35

Exercise 147

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?

Solution

72

Exercise 148

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?

Solution

72,000

Exercise 149

Find the fourth term of the expansion 2x3y82x3y8 size 12{ left (2x - 3y right ) rSup { size 8{8} } } {}.

Solution

48384x5y348384x5y3 size 12{ - "48384"x rSup { size 8{5} } y rSup { size 8{3} } } {}
(21)

Exercise 150

Find the coefficient of the a5b4a5b4 size 12{a rSup { size 8{5} } b rSup { size 8{4} } } {} term in the expansion of a2b9a2b9 size 12{ left (a - 2b right ) rSup { size 8{9} } } {}.

Solution

2016a5b42016a5b4 size 12{"2016"a rSup { size 8{5} } b rSup { size 8{4} } } {}
(22)

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