Question 22 Out of the 727 members of a club, 600 decide in favour of cricket while 173...
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Question 22
Out of the 727 members of a club, 600 decide in favour of cricket while 173 decide in favour of both cricket and hockey. Each member gives his opinion about at least one of the two games.
How many of them decide in favour of only hockey?
Show answer & explanation
Using the principle of set union: Total = Cricket + Hockey - Both. Since 600 favour cricket (including those who favour both), and 173 favour both, the number who favour only cricket is 600 - 173 = 427. With 727 total members, those who favour only hockey = 727 - 427 - 173 = 127.
Step-by-step Derivation:
Let C = set of members favouring cricket, H = set of members favouring hockey.
Given:
- |C ∪ H| = 727 (total members, each votes for at least one)
- |C| = 600 (members favouring cricket)
- |C ∩ H| = 173 (members favouring both)
Using the formula: |C ∪ H| = |C| + |H| - |C ∩ H|
727 = 600 + |H| - 173
727 = 427 + |H|
|H| = 300
Members favouring only hockey = |H| - |C ∩ H| = 300 - 173 = 127
Verification:
- Only cricket: 600 - 173 = 427
- Both: 173
- Only hockey: 127
- Total: 427 + 173 + 127 = 727 ✓