Question 21 Out of the 727 members of a club, 600 decide in favour of cricket while 173...
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Question 21
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 hockey?
Show answer & explanation
Using the inclusion-exclusion principle: |C ∪ H| = |C| + |H| - |C ∩ H|. We know the total is 727, cricket fans are 600, and both are 173. Solving: 727 = 600 + |H| - 173, which gives |H| = 300. Therefore, 300 members decide in favour of hockey.
Step-by-step Derivation:
Let C = set of cricket fans, H = set of hockey fans.
Given:
- Total members: |C ∪ H| = 727
- Cricket fans: |C| = 600
- Both cricket and hockey: |C ∩ H| = 173
Using inclusion-exclusion principle:
|C ∪ H| = |C| + |H| - |C ∩ H|
727 = 600 + |H| - 173
727 = 427 + |H|
|H| = 727 - 427
|H| = 300
Verification:
- Only cricket: 600 - 173 = 427
- Both: 173
- Only hockey: 300 - 173 = 127
- Total: 427 + 173 + 127 = 727 ✓