Question

A class of 30 students comprises of boys who can play cricket, hockey and/or football. 3 boys play only cricket, 3 boys play only hockey and 2 play only football. 4 boys could play all 3 games, while 11 could play football and cricket and 10 boys could play football and hockey. How many boys played cricket and hockey but not football?

• A. 1
• B. 3
• C. 4
• D. 5

Hint:

For three-set Venn diagram problems, fill the central intersection first, then subtract it from the pairwise intersections before using the total.

Answer 1

The Correct Option is D

Concept: Use a three-set Venn diagram. Let:
• Cricket = \(C\)
• Hockey = \(H\)
• Football = \(F\) Given: \[ \text{Only Cricket}=3 \] \[ \text{Only Hockey}=3 \] \[ \text{Only Football}=2 \] \[ C\cap H\cap F = 4 \] Also, \[ |C\cap F|=11 \] Therefore, \[ (C\cap F\text{ only})=11-4=7 \] Similarly, \[ |H\cap F|=10 \] Hence, \[ (H\cap F\text{ only})=10-4=6 \] Let \[ x=(C\cap H\text{ only}) \] Since total students are 30, \[ 3+3+2+7+6+4+x=30 \] \[ 25+x=30 \] \[ x=5 \] Therefore, \[ \boxed{5} \] Hence, the number of boys who played cricket and hockey but not football is 5.