GATE CS 2010 Q59 — General Aptitude

We can solve this problem using the principle of inclusion-exclusion for two sets.

Let:

• $U$ be the universal set of all persons in the room, so $n(U) = 25$.
• $H$ be the set of persons who play hockey, so $n(H) = 15$.
• $F$ be the set of persons who play football, so $n(F) = 17$.
• $H \cap F$ be the set of persons who play both sports, so $n(H \cap F) = 10$.

First, we find the number of persons who play at least one of the two sports, which is represented by $n(H \cup F)$:

The number of persons who play neither hockey nor football is the complement of this union within the universal set:

Therefore, the number of persons playing neither sport is 3.