This Exam P sample reference tests Inclusion-Exclusion. Three-set inclusion-exclusion counts 45 viewers in the first three groups; adding the disjoint group gives 63, so 37 watched none, choice B.
Treat the first three viewing groups as overlapping sets and the fourth group as disjoint. The overlaps among the first three groups must be resolved before the fourth group is added.
N(C)=34,N(N)=15,N(A)=10,N(H)=18
Model
Model
Three-set inclusion-exclusion adds the individual group counts, subtracts the three pairwise overlaps, and restores the triple overlap once.
N(C∪N∪A)=34+15+10−7−6−5+4=45
Compute
Compute
The first three groups contain 45 people. Adding the 18 people in the disjoint fourth group gives 63 represented people, so 37 of the 100 surveyed people watched none.
N((C∪N∪A∪H)c)=100−(45+18)=37
Answer
Answer
Exactly 37 surveyed viewers are outside all four groups.
37(B)
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CThe value 45 is the union count for the first three groups. It stops before adding the disjoint fourth group and before taking the complement within 100.
DThe value 55 is obtained by subtracting the first three-group union from 100. This omits the 18 members of the disjoint fourth group before taking the complement.
EThe value 82 is obtained by removing only the 18 members of the fourth group from 100. It ignores everyone in the first three groups.
Original practice · fully worked
Original variant: members outside four activity groups
A club has 100 members. Membership counts for hiking, cycling, and rowing are 40, 25, and 15; pairwise overlaps are 10, 8, and 5, and 3 join all three. A separate chess group has 12 members and none joins those activities. How many members join none of the four groups?
A 18
B 25
C 28
D 40
E 72
Variant answer in brief
Inclusion-exclusion gives 60 members in the three activity groups. The disjoint chess group raises the covered count to 72, leaving 28, choice C.
Setup
Setup
Treat the three activity groups as overlapping sets and the chess group as disjoint. First determine how many members belong to at least one activity group.
N(H)=40,N(C)=25,N(R)=15,N(S)=12
Model
Model
Three-set inclusion-exclusion adds the activity counts, subtracts every pairwise overlap, and adds the triple overlap back once.
N(H∪C∪R)=40+25+15−10−8−5+3=60
Compute
Compute
The activity union contains 60 members. Including the 12 chess members gives 72 covered members, leaving 28 outside all four groups.
N(none)=100−(60+12)=28
Answer
Answer
Twenty-eight members join none of the four groups.
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