Question

In a sports event of football and basketball, 132 students registered to play football and 93 students registered in basketball. If the total number of students registered in the event is 200, then the number of students registered in both the games is:

• A. 20
• B. 25
• C. 32
• D. 27

Hint:

The Principle of Inclusion-Exclusion is essential for problems involving overlapping sets. For two sets A and B, the size of their union is the sum of their individual sizes minus the size of their intersection: \(|A \cup B| = |A| + |B| - |A \cap B|\).

Answer 1

The Correct Option is B

Step 1: Define the sets and list the given information.
Let F be the set of students who registered for football. Let B be the set of students who registered for basketball. We are given: - Number of students in football, \( |F| = 132 \) - Number of students in basketball, \( |B| = 93 \) - Total number of students, which is the number of students in at least one of the games, \( |F \cup B| = 200 \).

Step 2: Use the Principle of Inclusion-Exclusion.
The formula for two sets is: \[ |F \cup B| = |F| + |B| - |F \cap B| \] We want to find the number of students registered in both games, which is \( |F \cap B| \).

Step 3: Substitute the given values and solve for \( |F \cap B| \). \[ 200 = 132 + 93 - |F \cap B| \] \[ 200 = 225 - |F \cap B| \] \[ |F \cap B| = 225 - 200 \] \[ |F \cap B| = 25 \] So, 25 students registered for both games.