Question

Let A, B, C denote the set of students in a college who play football, basketball, and cricket respectively. If \( n(A) = 60 \), \( n(B) = 55 \), \( n(C) = 70 \), \( n(A \cup B \cup C) = 100 \) and \( n(A \cap B \cap C) = 20 \), then the number of students who play exactly two of these sports is

• A. 40
• B. 45
• C. 60
• D. 75
• E. 85

Hint:

Use the principle of inclusion-exclusion to solve problems involving unions and intersections of sets.

Answer 1

The Correct Option is B

Using the principle of inclusion-exclusion, we can calculate the number of students who play exactly two sports: \[ n({exactly two}) = n(A \cap B) + n(B \cap C) + n(A \cap C) - 3n(A \cap B \cap C) \] From the problem: - \( n(A \cup B \cup C) = 100 \) - \( n(A) = 60 \), \( n(B) = 55 \), \( n(C) = 70 \) - \( n(A \cap B \cap C) = 20 \) Thus, the number of students who play exactly two sports is \( 45 \). 
Thus, the correct answer is (B).