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

$45$

$43$

$32$

$38$

✅ Correct Option: 2

We need to organize the information about club members and their sport preferences using a systematic approach. Let us solve this using a Venn diagram method with three overlapping circles.

We have 256 total members where:

144 play football
123 play tennis
132 play cricket
58 play both football and tennis
25 play both cricket and tennis
63 play both football and cricket
Everyone plays at least one game

Let us define $x$ = number of members who can play all three games.

Using the intersection information:

Members playing football and tennis (including those who play all three): 58
Members playing cricket and tennis (including those who play all three): 25
Members playing football and cricket (including those who play all three): 63

Key Insight: When we say "58 can play both football and tennis," this includes those who play all three games.

So:

Only football and tennis (not cricket) = $58 - x$
Only cricket and tennis (not football) = $25 - x$
Only football and cricket (not tennis) = $63 - x$

For only tennis:

Total tennis players = Only tennis + (Football & Tennis only) + (Cricket & Tennis only) + (All three)

$123 = \text{Only tennis} + (58 - x) + (25 - x) + x$

$123 = \text{Only tennis} + 58 + 25 - x$

Only tennis = $40 + x$

Similarly:

Only football = $23 + x$
Only cricket = $44 + x$

Total members = Only football + Only tennis + Only cricket + (F&T only) + (C&T only) + (F&C only) + All three

$256 = (23 + x) + (40 + x) + (44 + x) + (58 - x) + (25 - x) + (63 - x) + x$

$256 = 23 + 40 + 44 + 58 + 25 + 63 + x$

$256 = 253 + x$

$x = 3$

Members who can play only tennis = $40 + x = 40 + 3 = 43$

Important Note: This method ensures we don't double-count anyone and systematically accounts for all possible combinations of sports preferences.

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