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
Answer and calculation
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.
28(C) 