A club has 256 members of whom 144 can play football, 123 can play tennis, and…
2025
A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, 58 members can play both football and tennis, 25 can play both cricket and tennis, while 63 can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is
- A.
32
- B.
43
- C.
38
- D.
45
Show answer & explanation
Correct answer: B
Concept: For three sets whose union is the entire population (every member plays at least one game), the inclusion–exclusion principle gives |F ∪ T ∪ C| = |F| + |T| + |C| − |F ∩ T| − |T ∩ C| − |F ∩ C| + |F ∩ T ∩ C|. To count members in only one set, subtract from that set's total every pairwise overlap it takes part in, then add back the triple overlap once — because both pairwise subtractions removed it twice.
Application:
Given: total members (the union) = 256; n(F) = 144, n(T) = 123, n(C) = 132; n(F ∩ T) = 58, n(T ∩ C) = 25, n(F ∩ C) = 63.
Substitute into the union equation to isolate the triple intersection: 256 = 144 + 123 + 132 − 58 − 25 − 63 + n(F ∩ T ∩ C).
Simplify the right-hand side: 144 + 123 + 132 = 399, and 58 + 25 + 63 = 146, so 256 = 399 − 146 + n(F ∩ T ∩ C) = 253 + n(F ∩ T ∩ C).
Solve for the triple intersection: n(F ∩ T ∩ C) = 256 − 253 = 3.
Tennis-only count = n(T) − n(F ∩ T) − n(T ∩ C) + n(F ∩ T ∩ C) = 123 − 58 − 25 + 3 = 43.
Cross-check: Splitting the tennis circle by region — only tennis, tennis-and-football only (n(F ∩ T) − 3 = 55), tennis-and-cricket only (n(T ∩ C) − 3 = 22), and all three sports (3) — must add back up to n(T) = 123. Indeed 43 + 55 + 22 + 3 = 123, confirming the tennis-only count.