Deck 8: Counting and Probability

Question 1

Solve the problem.
Among a group of 84 investors, 25 owned shares of Stock A, 28 owned shares of Stock B, 43 owned shares of Stock C, 11 owned shares of both Stock A and Stock B, 11 owned shares of Stock A and Stock C, 16 owned shares
Of Stock B and Stock C, and 7 owned shares of all three. How many investors did not have shares in any of the
Three? How many owned shares of either Stock A or Stock C but not Stock B?

A) 19; 33
B) 26; 33
C) 19; 40
D) 19; 37

Answer 1

D

Question 2

Use the information given in the figure.
How many are not in C?

A) 39
B) 45
C) 36
D) 40

Answer 2

D

Question 3

Solve the problem.
In a survey of 159 vacationers in a popular beach resort town, 62 indicated they would consider buying a home there, 52 would consider buying a beach villa, 46 would consider buying a lot, 21 would consider both a home
And a villa, 20 would consider both a home and a lot, 18 would consider both a villa and a lot, and 9 would
Consider all three. How many vacationers would not consider any of the three? How many would consider only
A home?

A) 49; 17
B) 49; 30
C) 58; 42
D) 49; 22

Answer 3

B

Question 4

Solve the problem.
In survey of 50 households, 25 responded that they have an HDTV television, 35 responded that they had a multimedia personal computer and 15 responded they had both. How many households had neither an HDTV
Television nor a multimedia personal computer?

A) 25
B) 35
C) 5
D) 15

Question 5

Solve the problem.
In a survey of 53 hospital patients, 19 said they were satisfied with the nursing care, 25 said they were satisfied with the medical treatment, and 6 said they were satisfied with both. How many patients were satisfied with
Neither? How many were satisfied with only the medical treatment?

A) 15; 19
B) 15; 25
C) 13; 19
D) 21; 25

Question 6

Solve the problem.
The following data represent the marital status of females 18 years and older in a certain U.S. city. $$\begin{array} { l | c } \text { Marital Status } & \text { Number (in thousands) } \\\hline \text { Married } & 311 \\\text { Widowed } & 57 \\\text { Divorced } & 62 \\\text { Never married } & 111\end{array}$$ Determine the number of females 18 years old and older who are married or widowed.

A) 368,000
B) 311,000
C) 373,000
D) 430,000

Question 7

Use the information given in the figure.
How many are in A or B or C?

A) 11
B) 2
C) 81
D) 89

Question 8

Solve the problem.
In a student survey, 95 students indicated that they speak Spanish, 32 students indicated that they speak French, 13 students indicated that they speak both Spanish and French, and 139 students indicated that they speak
Neither. How many students participated in the survey?

A) 114
B) 240
C) 266
D) 253

Question 9

Solve the problem.
A survey of 2024 credit card users indicated that 827 had bought books online, 938 had bought music online, 448 had bought pet supplies online, 93 had bought both books and music, 193 had bought both books and pet
Supplies, 112 had bought both music and pet supplies, and 56 had bought all three. How many credit card users
Did not buy any of the three? How many bought either books or pet supplies but not music?

A) 153; 796
B) 153; 933
C) 209; 796
D) 153; 852

Question 10

Use the information given in the figure.

How many are in set A?

A) 34
B) 15
C) 23
D) 20

Question 11

Write down all the subsets of the given set.
$\{ 2,4,10,11 \}$

A) $\{ 2 \} , \{ 4 \} , \{ 10 \} , \{ 11 \} , \{ 2,4 \} , \{ 2,10 \} , \{ 2,11 \} , \{ 4,10 \}$ ,
$\{ 4,11 \} , \{ 10,11 \} , \{ 2,4,10 \} , \{ 2,4,11 \} , \{ 2,10,11 \}$ , $\{ 4,10,11 \} , \varnothing$
B) $\{ 2 \} , \{ 4 \} , \{ 10 \} , \{ 11 \} , \{ 2,4 \} , \{ 2,10 \} , \{ 2,11 \} , \{ 4,10 \}$ , $\{ 4,11 \} , \{ 10,11 \} , \{ 2,4,10 \} , \{ 2,4,11 \} , \{ 2,10,11 \} , \{ 4,10,11 \} , \{ 2,4,10,11 \} , \varnothing$
C) $\{ 2 \} , \{ 4 \} , \{ 10 \} , \{ 11 \} , \{ 2,4 \} , \{ 2,10 \} , \{ 2,11 \} , \{ 4,10 \}$ , $\{ 4,11 \} , \{ 2,4,10 \} , \{ 2,4,11 \} , \{ 2,10,11 \} , \{ 4,10,11 \}$ , $\{ 2,4,10,11 \} , \varnothing$
D) $\{ 2 \} , \{ 4 \} , \{ 10 \} , \{ 11 \} , \{ 2,4 \} , \{ 2,10 \} , \{ 2,11 \} , \{ 4,10 \}$ , $\{ 4,11 \} , \{ 10,11 \} , \{ 2,4,10 \} , \{ 2,4,11 \} , \{ 2,10,11 \} , \{ 4,10,11 \} , \{ 2,4,10,11 \}$

Question 12

Write down all the subsets of the given set.
$\{ p , q , r \}$

A) $\varnothing , \{ p \} , \{ q \} , \{ r \} , \{ p , q \} , \{ p , r \} , \{ q , r \} , \{ p , q , r \}$
B) $\varnothing , \{ p \} , \{ q \} , \{ r \} , \{ p , q \} , \{ p , r \} , \{ q , r \}$
C) $\varnothing , \{ p \} , \{ q \} , \{ r \} , \{ p , q \} , \{ p , r \} , \{ q , r \} , \{ p , p \} , \{ q , q \} , \{ r , r \} , \{ p , q , r \}$
D) $\{ p \} , \{ q \} , \{ r \} , \{ p , q \} , \{ p , r \} , \{ q , r \} , \{ p , q , r \}$

Question 13

Solve the problem.
In a survey of 415 computer buyers, 185 put price as a main consideration, 236 put performance as a main consideration, and 57 listed both price and performance. How many computer buyers listed other
Considerations? How many looked only for performance?

A) 51; 236
B) 128; 179
C) 51; 179
D) 108; 236

Question 14

Use the information given in the figure.
How many are in A and B and C?

A) 58
B) 71
C) 1
D) 13

Question 15

Use the information given in the figure.

How many are in B but not in A?

A) 22
B) 27
C) 32
D) 33

Question 16

Solve the problem.
A man has 4 shirts and 7 ties. How many different shirt and tie arrangements can he wear?

A) 16
B) 28
C) 56
D) 49

Question 17

Write down all the subsets of the given set.
{a}

A) {a, b}
B) a
C) {a}
D) ∅, {a}

Question 18

Use the information given in the figure.

How many are in B or C?

A) 48
B) 52
C) 5
D) 54

Question 19

Write down all the subsets of the given set.
$\{ 4 , \alpha , 9 , \pi \}$

A) $\{ 4 \} , \{ \alpha \} , \{ 9 \} , \{ \pi \} , \{ 4 , \alpha \} , \{ 4,9 \} , \{ 4 , \pi \} , \{ \alpha , 9 \}$ , $\{ \alpha , \pi \} , \{ 9 , \pi \} , \{ 4 , \alpha , 9 \} , \{ 4 , \alpha , \pi \} , \{ 4,9 , \pi \} , \{ \alpha , 9 , \pi \} , \varnothing$
B) $\{ 4 \} , \{ \alpha \} , \{ 9 \} , \{ \pi \} , \{ 4 , \alpha \} , \{ 4,9 \} , \{ 4 , \pi \} , \{ \alpha , 9 \}$ , $\{ \alpha , \pi \} , \{ 9 , \pi \} , \{ 4 , \alpha , \pi \} , \{ 4,9 , \pi \} , \{ \alpha , 9 , \pi \}$ , $\{ 4 , \alpha , 9 , \pi \} , \varnothing$
C) $\{ 4 \} , \{ \alpha \} , \{ 9 \} , \{ \pi \} , \{ 4 , \alpha \} , \{ 4,9 \} , \{ 4 , \pi \} , \{ \alpha , 9 \}$ , $\{ \alpha , \pi \} , \{ 9 , \pi \} , \{ 4 , \alpha , 9 \} , \{ 4 , \alpha , \pi \} , \{ 4,9 , \pi \} , \{ \alpha , 9 , \pi \}$ , $\{ 4 , \alpha , 9 , \pi \} , \varnothing$
D) $\{ 4 \} , \{ \alpha \} , \{ 9 \} , \{ \pi \} , \{ 4 , \alpha \} , \{ 4,9 \} , \{ 4 , \pi \} , \{ \alpha , 9 \}$ , $\{ \alpha , \pi \} , \{ 9 , \pi \} , \{ 4 , \alpha , 9 \} , \{ 4 , \alpha , \pi \} , \{ 4,9 , \pi \} , \{ \alpha , 9 , \pi \}$ , $\{ 4 , \alpha , 9 , \pi \}$

Question 20

Use the information given in the figure.

How many are in B and C?

A) 1
B) 37
C) 3
D) 47

Question 21

Solve the problem.
How many arrangements of answers are possible in a multiple-choice test with 7 questions, each of which has 3 possible answers?

A) 35
B) 840
C) 2187
D) 210

Question 22

Solve the problem.
A certain mathematics test consists of 20 questions. Goldie decides to answer the questions without reading them. In how many ways can Goldie fill in the answer sheet if the possible answers are true and false?

A) 1,048,576
B) 190
C) 40
D) 400

Question 23

Find the value of the permutation.
P(9, 0)

A) 5
B) 1
C) 362,880
D) 725,760

Question 24

Solve the problem.
How many different license plates can be made using 3 letters followed by 2 digits selected from the digits 0 through 9, if neither letters nor digits may be repeated?

A) 1,404,000
B) 117,000
C) 1,757,600
D) 1,123,200

Question 25

Solve the problem.
In how many ways can 6 people be lined up?

A) 1
B) 6
C) 360
D) 720

Question 26

Solve the problem.
How many 6-symbol codes can be formed using 4 different symbols? Repeated symbols are allowed.

A) 15
B) 30
C) 360
D) 4096

Question 27

Find the value of the permutation.
P(7, 7)

A) 2
B) 2520
C) 1
D) 5040

Question 28

Solve the problem.
How many 5-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0? Repeated digits are allowed.

A) 59,049
B) 15,120
C) 90,000
D) 45,360

Question 29

Solve the problem.
How many 2-letter codes can be formed using the letters A, B, C, D, E, F, G, H, and I. Repeated letters are allowed.

A) 512
B) 72
C) 36
D) 81

Question 30

Solve the problem.
A student must choose 1 of 5 mathematics electives, 1 of 5 science electives, and 1 of 8 programming electives. How many possible course selections are there?

A) 25 course selections
B) 400 course selections
C) 200 course selections
D) 18 course selections

Question 31

Solve the problem.
Lisa has 4 skirts, 9 blouses, and 3 jackets. How many 3-piece outfits can she put together assuming any piece goes with any other?

A) 216 possible outfits
B) 108 possible outfits
C) 36 possible outfits
D) 16 possible outfits

Question 32

Solve the problem.
A restaurant offers a choice of 4 salads, 7 main courses, and 4 desserts. How many possible 3-course meals are there?

A) 15 possible meals
B) 224 possible meals
C) 112 possible meals
D) 28 possible meals

Question 33

Find the value of the permutation.
P(7, 1)

A) 5040
B) 720
C) 7
D) 1

Question 34

Solve the problem.
12 different books are to be arranged on a shelf. How many different arrangements are possible?

A) 239,500,800
B) 12
C) 39,916,800
D) 479,001,600

Question 35

Solve the problem.
How many different license plates can be made using 2 letters followed by 2 digits selected from the digits 0 through 9, if letters and digits may be repeated?

A) 260
B) 67,600
C) 4
D) 36

Question 36

Find the value of the permutation.
P(10, 3)

A) 604,800
B) 360
C) 10,080
D) 720

Question 37

Solve the problem.
How many 2-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? No digit can be used more than once.

A) 90
B) 1,814,400
C) 3,628,800
D) 45

Question 38

Solve the problem.
List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choosing 3 at a time without repetition. What is P(4,3)? A) $123,124,134,234$
$\mathrm { P } ( 4,3 ) = 4$
B) $123,124,132,142,143,213,214,231,241,243,312,314,321,341,342,412,413,421,423,431$ $\mathrm { P } ( 4,3 ) = 20$
C) $111,112,113,114,121,122,123,124,131,132,133,134,141,142,143,144,211,212,213,214,221,222,223$ , $224,231,232,233,234,241,242,243,244,311,312,313,314,321,322,323,324,331,332,333,334,341,342$ , $343,344,411,412,413,414,421,422,423,424,431,432,433,434,441,442,443,444$ $\mathrm { P } ( 4,3 ) = 64$
D) $123,124,132,134,142,143,213,214,231,234,241,243,312,314,321,324,341,342,412,413,421,423,431$ , 432 $\mathrm { P } ( 4,3 ) = 24$

Question 39

Solve the problem.
List all the ordered arrangements of 6 objects a, b, c, d, e, and f choosing 2 at a time without repetition. What is P(6, 2)? A) aa, ab, ac, ad, ae, af, ba, bb, bc, bd, be, bf, ca, cb, cc, cd, ce, cf, da, db, dc, dd, de, df, ea, eb, ec, ed, ee, ef, fa, $\mathrm { fb } , \mathrm { fc } , \mathrm { fd } , \mathrm { fe } , \mathrm { ff }$
$\mathrm { P } ( 6,2 ) = 36$
B) ab, ac, ad, ae, af, ba, bc, bd, be, bf, ca, cb, cd, ce, cf, da, db, dc, de, df, ea, eb, ec, ed, ef
$\mathrm { P } ( 6,2 ) = 25$
C) ab, ac, ad, ae, af, ba, bc, bd, be, bf, ca, cb, cd, ce, cf, da, db, dc, de, df, ea, eb, ec, ed, ef, fa, fb, fc, fd, fe
$\mathrm { P } ( 6,2 ) = 30$
D) ab, ac, ad, ae, af, bc, bd, be, bf, cd, ce, cf, de, df, ef
$\mathrm { P } ( 6,2 ) = 15$

Question 40

Solve the problem.
How many different 7-letter codes are there if only the letters A, B, C, D, E, F, G, H, and I can be used and no letter can be used more than once?

A) 4,782,969
B) 36
C) 7
D) 181,440

Question 41

Solve the problem.
A group of 12 friends goes bowling. How many different possibilities are there for the order in which they play if the youngest person is to bowl first?

A) 39,916,800
B) 479,001,600
C) 12
D) 11

Question 42

Solve the problem.
An environmental organization has 20 members. Each member will be placed on exactly 1 of 4 teams. Each team will work on a different issue. The first team has 5 members, the second has 4, the third has 8, and the
Fourth has 3. In how many ways can these teams be formed?

A) 2.432902008 × 1018
B) 3.491888400 × 109
C) 1.078721965 × 1016
D) 1.047566520 × 1010

Question 43

Solve the problem.
List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is C(4, 3)? A) $123,124,134,234,321,432$
$$C ( 4,3 ) = 6$$
B) $123,124,134,234$
$$\mathrm { C } ( 4,3 ) = 4$$
C) $123,124,132,134,142,143,213,214,231,234,241,243,312,314,321,324,341,342,412,413,421,423,431$ , 432
$$\mathrm { C } ( 4,3 ) = 24$$
D) $111,112,113,114,121,122,123,124,131,132,133,134,141,142,143,144,211,212,213,214,221,222,223$ , $224,231,232,233,234,241,242,243,244,311,312,313,314,321,322,323,324,331,332,333,334,341,342$ , $343,344,411,412,413,414,421,422,423,424,431,432,433,434,441,442,443,444$ $C ( 4,3 ) = 64$

Question 44

Solve the problem.
A hot dog stand sells hot dogs with cheese, relish, chili, tomato, onion, mustard, or ketchup. How many different hot dogs can be concocted using any 5 of the extras?

A) 21
B) 1260
C) 42
D) 2520

Question 45

Find the value of the combination.
C(6, 6)

A) 180
B) 1
C) 720
D) 0.5

Question 46

Solve the problem.
In a probability model, which of the following numbers could be the probability of an outcome: $0,0.2 , - 0.01 , - \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 5 } { 4 } , 1,1.5$

A) $0.2 , \frac { 1 } { 2 } , 1$
B) $0,0.2 , - 0.01,1,1$
C) $0,0.2 , - 0.01 , - \frac { 1 } { 3 } , \frac { 1 } { 2 } , 1$
D) $0,0.2 , \frac { 1 } { 2 } , 1$

Question 47

Solve the problem.
In how many ways can 7 people each have different birth months?

A) 84
B) 3,991,680
C) 35,831,808
D) 792

Question 48

Determine whether the following is a probability model.
$$\begin{array} { l | c } \text { Outcome } & \text { Probability } \\\hline \text { Red } & 0.21 \\\text { Blue } & 0.23 \\\text { Green } & 0.24 \\\text { White } & 0.32\end{array}$$

A) Yes
B) No

Question 49

Solve the problem.
Mary finds 10 fish at a pet store that she would like to buy, but she can afford only 4 of them. In how many ways can she make her selection? How many ways can she make her selection if he decides that one of the fish
Is a must?

A) 210; 84
B) 151,200; 60,480
C) 5040; 504
D) 2520; 252

Question 50

Solve the problem.
How many different vertical arrangements are there of 8 flags if 4 are white, 2 are blue, and 2 are red?

A) 420
B) 20
C) 70
D) 16

Question 51

Solve the problem.
An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 6 of the multiple-choice questions and 4 of the essay questions, in how many ways can the questions be chosen?

A) 1296
B) 1260
C) 21,772,800
D) 261,273,600

Question 52

Solve the problem.
A committee is to be formed consisting of 2 men and 5 women. If the committee members are to be chosen from 9 men and 12 women, how many different committees are possible?

A) 6,842,880
B) 28,512
C) 828
D) 116,280

Question 53

Solve the problem.
How many ways are there to choose a soccer team consisting of 3 forwards, 4 midfield players, and 3 defensive players, if the players are chosen from 8 forwards, 10 midfield players, and 8 defensive players?

A) 322
B) 568,995,840
C) 658,560
D) 5,311,735

Question 54

Solve the problem.
How many different license plates can be made using 4 letters followed by 2 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

A) 41,127,840
B) 889.880952
C) 35,880,000
D) 45,697,600

Question 55

Solve the problem.
How many different 11-letter words (real or imaginary) can be formed from the letters of the word
MISSISSIPPI? Leave your answer in factorial form.

Question 56

Solve the problem.
From 9 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible?

A) 15,120
B) 126
C) 3024
D) 1512

Question 57

Find the value of the combination.
C(10, 6)

A) 48
B) 75,600
C) 5040
D) 210

Question 58

Solve the problem.
How many different 10-letter words (real or imaginary) can be formed from the letters in the word IMMUNOLOGY?

A) 907,200
B) 90,720
C) 1,814,400
D) 3,628,800

Question 59

Solve the problem.
List all the combinations of 6 objects a, b, c, d, e, and f taken 2 at a time. What is C(6, 2)? A) ab, ac, ad, ae, af, bc, bd, be, bf, cd, ce, cf, de, df, ef $\mathrm { C } ( 6,2 ) = 15$
B) ab, ac, ad, ae, af, ba, bc, bd, be, bf, ca, cb, cd, ce, cf, da, db, dc, de, df, ea, eb, ec, ed, ef, fa, fb, fc, fd, fe $C ( 6,2 ) = 30$
C) ab, ac, ad, ae, bc, bd, be, cd, ce, cf, de, df $\mathrm { C } ( 6,2 ) = 12$
D) aa, ab, ac, ad, ae, af, ba, bb, bc, bd, be, bf, ca, cb, cc, cd, ce, cf, da, db, dc, dd, de, df, ea, eb, ec, ed, ee, ef, fa, $\mathrm { fb } , \mathrm { fc } , \mathrm { fd } , \mathrm { fe } , \mathrm { ff }$ $C ( 6,2 ) = 36$

Question 60

Solve the problem.
How many 5-card poker hands consisting of three 5's and two cards that are not 5's are possible in a 52-card deck?

A) 2652
B) 5304
C) 4512
D) 2256

Question 61

Determine whether the following is a probability model.
$$\begin{array} { l | c } \text { Outcome } & \text { Probability } \\\hline \text { Golfing } & 0.13 \\\text { Skiing } & 0.15 \\\text { Swimming } & 0.19 \\\text { Biking } & 0.30 \\\text { Hiking } & 0.23\end{array}$$

A) Yes
B) No

Question 62

Determine whether the following is a probability model.
$$\begin{array} { l | c } \text { Outcome } & \text { Probability } \\\hline \text { Red } & 0.16 \\\text { Blue } & 0.19 \\\text { Green } & 0.27 \\\text { White } & 0.23\end{array}$$

A) Yes
B) No

Question 63

Determine whether the following is a probability model.
$$\begin{array} { l | c } \text { Outcome } & \text { Probability } \\\hline \text { Jim } & 0 \\\text { Tom } & 0 \\\text { Bill } & 1 \\\text { Carl } & 0\end{array}$$

A) Yes
B) No

Question 64

Determine whether the following is a probability model.
$$\begin{array} { l | c } \text { Outcome } & \text { Probability } \\\hline \text { Red } & - 0.23 \\\text { Blue } & 0.28 \\\text { Green } & 0.32 \\\text { White } & 0.17\end{array}$$

A) Yes
B) No

Question 65

Determine whether the following is a probability model.

A) Yes
B) No