Question:
There are 3 clubs A, B \& C in a town with 40, 50 \& 60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?
There are 3 clubs A, B \& C in a town with 40, 50 \& 60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?
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Use set theory and inclusion counts to handle overlaps in membership problems. Carefully track multiple-counting.
Updated On: Aug 7, 2025
- 20
- 25
- 50
- 70
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The Correct Option is B
Solution and Explanation
Let:
- Total in A = 40, B = 50, C = 60
- People in all 3 = 10
- Only one club = 70
- Total members = $|A| + |B| + |C| = 150$
(Note: total counts duplicate people)
Let:
- Let $x$ = number in exactly two clubs
- Let $z = 10$ = number in all three
- Let $y = 70$ = number in only one club
Total distinct people = $y + x + z$
\[
= 70 + x + 10 = 80 + x
\]
But total member count from all three clubs = 150
Each person in exactly two clubs is counted twice → total contribution = $2x$
Each in all three is counted thrice → $3z = 30$
Each in only one is counted once → $70$
So:
\[
70 + 2x + 30 = 150 \Rightarrow 2x = 50 \Rightarrow x = \boxed{25}
\]
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