Question

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. 48
• B. 43
• C. 45
• D. 52
• E. 50

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is a set theory problem involving three sets (Football, Tennis, Cricket). We need to find the size of the region "Tennis only".
Step 2: Key Formula or Approach:
Use the Principle of Inclusion-Exclusion:
\[ n(F \cup T \cup C) = n(F) + n(T) + n(C) - n(F \cap T) - n(T \cap C) - n(F \cap C) + n(F \cap T \cap C) \]
Step 3: Detailed Explanation:

Knowns:
- \(n(Total) = 256\)
- \(n(F) = 144, n(T) = 123, n(C) = 132\)
- \(n(F \cap T) = 58, n(T \cap C) = 25, n(F \cap C) = 63\)

Step 1: Find people playing all three (\(x\)):
\(256 = 144 + 123 + 132 - (58 + 25 + 63) + x\)
\(256 = 399 - 146 + x\)
\(256 = 253 + x \rightarrow x = 3\)

Step 2: Find "Only Tennis":
People in the Tennis circle are made of:
Only Tennis + (Tennis \(\cap\) Football only) + (Tennis \(\cap\) Cricket only) + (All 3).
- (Tennis \(\cap\) Football only) = \(58 - 3 = 55\)
- (Tennis \(\cap\) Cricket only) = \(25 - 3 = 22\)
- (All 3) = \(3\)
Only Tennis = \(123 - (55 + 22 + 3)\)
Only Tennis = \(123 - 80 = 43\).

Step 4: Final Answer:
The number of members who can play only tennis is 43. Option (B) is correct.