Ch. 14 Counting and Probability
14.1 Counting
1 Find All the Subsets of a Set
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write down all the subsets of the given set.
1) {3, 5, 6, 7}
A) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {6, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {5, 6, 7}, {3, 5, 6, 7}
B) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {6, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7},
{5, 6, 7}, ∅
C) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {5, 6, 7},
{3, 5, 6, 7}, ∅
D) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {6, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {5, 6, 7}, {3, 5, 6, 7}, ∅
2) {1, α, 10, π}
A) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, 10}, {1, α, π}, {1, 10, π}, {α, 10, π},
{1, α, 10, π}
B) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, π}, {1, 10, π}, {α, 10, π},
{1, α, 10, π}, ∅
C) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, 10}, {1, α, π}, {1, 10, π}, {α, 10, π},
{1, α, 10, π}, ∅
D) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, 10}, {1, α, π}, {1, 10, π}, {α, 10, π}, ∅
3) {a}
A) {a, b} B) {a} C) a D) ∅, {a}
4) {p, q, r}
A) ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}
B) ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, p}, {q, q}, {r, r}, {p, q, r}
C) ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}
D) {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}
2 Count the Number of Elements in a Set
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If n(A) = 55, n(B) = 49, and n(A ∩ B) = 17, find n(A ∪ B).
A) 87 B) 121 C) 104 D) 70
2) If n(A) = 58, n(B) = 55, and n(A ∪ B) = 94, find n(A ∩ B).
A) 38 B) 113 C) 19 D) 75
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, 3) If n(B) = 12, n(A ∩ B) = 3, and n(A ∪ B) = 21, find n(A).
A) 10 B) 14 C) 12 D) 9
4) If n(A) = 10, n(A ∪ B) = 28, and n(A ∩ B) = 6, find n(B).
A) 24 B) 18 C) 23 D) 25
5) If n(A ∪ B) = 52, n(A ∩ B) = 30, and n(A) = n(B), find n(A).
A) 41 B) 15 C) 11 D) 26
Use the information given in the figure.
6)
4
27 2 18
5 2
24 9
How many are in set A?
A) 38 B) 27 C) 47 D) 36
7)
3
26 4 30
1 3
22 3
How many are in B or C?
A) 7 B) 63 C) 59 D) 55
8)
2
22 5 23
5 3
16 3
How many are in B and C?
A) 42 B) 5 C) 8 D) 54
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, 9)
5
30 4 29
5 4
22 5
How many are in B but not in A?
A) 37 B) 42 C) 29 D) 33
10)
3
28 2 17
1 4
30 6
How many are not in C?
A) 54 B) 55 C) 48 D) 51
11)
3
27 1 21
2 1
15 3
How many are in A and B and C?
A) 7 B) 70 C) 1 D) 63
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, 12)
2
30 5 19
4 3
22 7
How many are in A or B or C?
A) 92 B) 14 C) 5 D) 85
Solve the problem.
13) In a survey of 49 hospital patients, 25 said they were satisfied with the nursing care, 19 said they were satisfied
with the medical treatment, and 5 said they were satisfied with both. How many patients were satisfied with
neither? How many were satisfied with only the medical treatment?
A) 10; 14 B) 10; 19 C) 20; 14 D) 15; 19
14) In a survey of 363 computer buyers, 167 put price as a main consideration, 204 put performance as a main
consideration, and 60 listed both price and performance. How many computer buyers listed other
considerations? How many looked only for performance?
A) 52; 204 B) 52; 144 C) 112; 204 D) 107; 144
15) 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) 35 B) 15 C) 25 D) 5
16) In a student survey, 109 students indicated that they speak Spanish, 34 students indicated that they speak
French, 10 students indicated that they speak both Spanish and French, and 140 students indicated that they
speak neither. How many students participated in the survey?
A) 133 B) 273 C) 263 D) 283
17) Among a group of 77 investors, 25 owned shares of Stock A, 30 owned shares of Stock B, 40 owned shares of
Stock C, 11 owned shares of both Stock A and Stock B, 13 owned shares of Stock A and Stock C, 17 owned
shares of Stock B and Stock C, and 9 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) 14; 29 B) 23; 29 C) 14; 38 D) 14; 33
18) In a survey of 166 vacationers in a popular beach resort town, 67 indicated they would consider buying a home
there, 72 would consider buying a beach villa, 55 would consider buying a lot, 16 would consider both a home
and a villa, 24 would consider both a home and a lot, 19 would consider both a villa and a lot, and 6 would
consider all three. How many vacationers would not consider any of the three? How many would consider
only a home?
A) 25; 43 B) 25; 33 C) 31; 43 D) 25; 18
19) A survey of 2125 credit card users indicated that 899 had bought books online, 1005 had bought music online,
383 had bought pet supplies online, 91 had bought both books and music, 170 had bought both books and pet
supplies, 158 had bought both music and pet supplies, and 64 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) 193; 821 B) 193; 927 C) 193; 885 D) 257; 821
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14.1 Counting
1 Find All the Subsets of a Set
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write down all the subsets of the given set.
1) {3, 5, 6, 7}
A) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {6, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {5, 6, 7}, {3, 5, 6, 7}
B) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {6, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7},
{5, 6, 7}, ∅
C) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {5, 6, 7},
{3, 5, 6, 7}, ∅
D) {3}, {5}, {6}, {7}, {3, 5}, {3, 6}, {3, 7}, {5, 6},
{5, 7}, {6, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {5, 6, 7}, {3, 5, 6, 7}, ∅
2) {1, α, 10, π}
A) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, 10}, {1, α, π}, {1, 10, π}, {α, 10, π},
{1, α, 10, π}
B) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, π}, {1, 10, π}, {α, 10, π},
{1, α, 10, π}, ∅
C) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, 10}, {1, α, π}, {1, 10, π}, {α, 10, π},
{1, α, 10, π}, ∅
D) {1}, {α}, {10}, {π}, {1, α}, {1, 10}, {1, π}, {α, 10},
{α, π}, {10, π}, {1, α, 10}, {1, α, π}, {1, 10, π}, {α, 10, π}, ∅
3) {a}
A) {a, b} B) {a} C) a D) ∅, {a}
4) {p, q, r}
A) ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}
B) ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, p}, {q, q}, {r, r}, {p, q, r}
C) ∅, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}
D) {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}
2 Count the Number of Elements in a Set
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If n(A) = 55, n(B) = 49, and n(A ∩ B) = 17, find n(A ∪ B).
A) 87 B) 121 C) 104 D) 70
2) If n(A) = 58, n(B) = 55, and n(A ∪ B) = 94, find n(A ∩ B).
A) 38 B) 113 C) 19 D) 75
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, 3) If n(B) = 12, n(A ∩ B) = 3, and n(A ∪ B) = 21, find n(A).
A) 10 B) 14 C) 12 D) 9
4) If n(A) = 10, n(A ∪ B) = 28, and n(A ∩ B) = 6, find n(B).
A) 24 B) 18 C) 23 D) 25
5) If n(A ∪ B) = 52, n(A ∩ B) = 30, and n(A) = n(B), find n(A).
A) 41 B) 15 C) 11 D) 26
Use the information given in the figure.
6)
4
27 2 18
5 2
24 9
How many are in set A?
A) 38 B) 27 C) 47 D) 36
7)
3
26 4 30
1 3
22 3
How many are in B or C?
A) 7 B) 63 C) 59 D) 55
8)
2
22 5 23
5 3
16 3
How many are in B and C?
A) 42 B) 5 C) 8 D) 54
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, 9)
5
30 4 29
5 4
22 5
How many are in B but not in A?
A) 37 B) 42 C) 29 D) 33
10)
3
28 2 17
1 4
30 6
How many are not in C?
A) 54 B) 55 C) 48 D) 51
11)
3
27 1 21
2 1
15 3
How many are in A and B and C?
A) 7 B) 70 C) 1 D) 63
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, 12)
2
30 5 19
4 3
22 7
How many are in A or B or C?
A) 92 B) 14 C) 5 D) 85
Solve the problem.
13) In a survey of 49 hospital patients, 25 said they were satisfied with the nursing care, 19 said they were satisfied
with the medical treatment, and 5 said they were satisfied with both. How many patients were satisfied with
neither? How many were satisfied with only the medical treatment?
A) 10; 14 B) 10; 19 C) 20; 14 D) 15; 19
14) In a survey of 363 computer buyers, 167 put price as a main consideration, 204 put performance as a main
consideration, and 60 listed both price and performance. How many computer buyers listed other
considerations? How many looked only for performance?
A) 52; 204 B) 52; 144 C) 112; 204 D) 107; 144
15) 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) 35 B) 15 C) 25 D) 5
16) In a student survey, 109 students indicated that they speak Spanish, 34 students indicated that they speak
French, 10 students indicated that they speak both Spanish and French, and 140 students indicated that they
speak neither. How many students participated in the survey?
A) 133 B) 273 C) 263 D) 283
17) Among a group of 77 investors, 25 owned shares of Stock A, 30 owned shares of Stock B, 40 owned shares of
Stock C, 11 owned shares of both Stock A and Stock B, 13 owned shares of Stock A and Stock C, 17 owned
shares of Stock B and Stock C, and 9 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) 14; 29 B) 23; 29 C) 14; 38 D) 14; 33
18) In a survey of 166 vacationers in a popular beach resort town, 67 indicated they would consider buying a home
there, 72 would consider buying a beach villa, 55 would consider buying a lot, 16 would consider both a home
and a villa, 24 would consider both a home and a lot, 19 would consider both a villa and a lot, and 6 would
consider all three. How many vacationers would not consider any of the three? How many would consider
only a home?
A) 25; 43 B) 25; 33 C) 31; 43 D) 25; 18
19) A survey of 2125 credit card users indicated that 899 had bought books online, 1005 had bought music online,
383 had bought pet supplies online, 91 had bought both books and music, 170 had bought both books and pet
supplies, 158 had bought both music and pet supplies, and 64 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) 193; 821 B) 193; 927 C) 193; 885 D) 257; 821
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