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 sets and known values.
Let F represent students in football, and B represent students in basketball. We know: \( |F| = 132 \), \( |B| = 93 \), and \( |F \cup B| = 200 \).

Step 2: Apply the Inclusion-Exclusion Principle.
For two sets, the formula is: \[ |F \cup B| = |F| + |B| - |F \cap B| \]. Our goal is to determine \( |F \cap B| \), the number of students in both football and basketball.

Step 3: Calculate \( |F \cap B| \).Substituting the given values: \[ 200 = 132 + 93 - |F \cap B| \].Simplifying: \[ 200 = 225 - |F \cap B| \].Therefore: \[ |F \cap B| = 225 - 200 \], which gives \[ |F \cap B| = 25 \]. Thus, 25 students are registered in both games.