Q.In a class of 60 students, 25 students play cricket and 20 students play tennis, and 10 students play both the games. Then, the number of students who play neither is
(A) $0$
(B) $25$
(C) $35$
(D) $45$

Concept: Set Operations (Inclusion-Exclusion Principle)

We need to find how many students play at least one game, then subtract from the total.

Let $C$ = set of cricket players, $T$ = set of tennis players.

Given: $|C| = 25$, $|T| = 20$, $|C \cap T| = 10$, and total students = 60.

By inclusion-exclusion:

So 35 students play at least one game. …

When we have overlapping sets, the natural instinct is to add the sizes — but that double-counts the overlap. The inclusion-exclusion principle corrects this: for two sets $A$ and $B$, the number in at least one is $|A| + |B| - |A \cap B|$. Once we know how many play at least one game, the remainder play neither.

Let's denote:

• $C$ = set of students who play cricket, so $|C| = 25$
• $T$ = set of students who play tennis, so $|T| = 20$
• $|C \cap T| = 10$ (students playing both)
• Total students = $60$

We want to find the number who play neither, which is the complement of "at least one game."

Step-by-step solution

1. Find the number playing at least one game By inclusion-exclusion:

Substituting the values:

So $35$ students play at least one of the two games (cricket or tennis or both).

2. Find the number playing neither The students who play neither are those not in $C \cup T$: $\text{Neither} = \text{Total} - |C \cup T| = 60 - 35 = 25$ …