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?

• A. 20
• B. 25
• C. 50
• D. 70

Hint:

Use set theory and inclusion counts to handle overlaps in membership problems. Carefully track multiple-counting.

Answer 1

The Correct Option is B

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: \[ 70 + 2x + 30 = 150 \Rightarrow 2x = 50 \Rightarrow x = \boxed{25} \]