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. 45
• B. 38
• C. 32
• D. 43

Answer 1

The Correct Option is D

To find the number of members who can play only tennis, we apply the principle of inclusion-exclusion to calculate the number of members who can play any of the games. Let:

• \(F = \) number of members who can play football = 144
• \(T = \) number of members who can play tennis = 123 
• \(C = \) number of members who can play cricket = 132
• \(FT = \) number of members who can play both football and tennis = 58
• \(TC = \) number of members who can play both tennis and cricket = 25
• \(FC = \) number of members who can play both football and cricket = 63
• Total members = 256

Applying the principle of inclusion-exclusion:
\( N(F \cup T \cup C) = F + T + C - FT - TC - FC + N(F \cap T \cap C) \)

 

where \(N(F \cap T \cap C)\) is the number of members who can play all three games. Every member plays at least one game, so \( N(F \cup T \cup C) = 256 \).
Substitute the known values:\( 256 = 144 + 123 + 132 - 58 - 25 - 63 + N(F \cap T \cap C) \)

 

\( 256 = 339 - 146 + N(F \cap T \cap C) \)

 

\( 256 = 193 + N(F \cap T \cap C) \)

 

\( N(F \cap T \cap C) = 256 - 193 = 63 \)

 

Now, calculate the number of members who can play only tennis:

• Only tennis = \( T - FT - TC + N(F \cap T \cap C) = 123 - 58 - 25 + 63 = 43 \)

Therefore, the number of members who can play only tennis is 43.