Question

In a class of 50 students, 30 play football, 25 play cricket, and 10 play both. How many play neither?

• A. 5
• B. 10
• C. 15
• D. 20

Hint:

Use inclusion–exclusion principle for set problems.

Answer 1

The Correct Option is A

To solve this problem, we need to determine how many students play neither football nor cricket. This involves using the principle of inclusion and exclusion.

Let:

• \(F\) be the set of students who play football.
• \(C\) be the set of students who play cricket.

We're given:

• \(|F| = 30\) (students play football)
• \(|C| = 25\) (students play cricket)
• \(|F \cap C| = 10\) (students play both football and cricket)
• \(|U| = 50\) (total students in the class)

To find the students who play either football or cricket or both, we use:

\[ |F \cup C| = |F| + |C| - |F \cap C| \]

Substituting the values:

\[ |F \cup C| = 30 + 25 - 10 = 45 \]

This means 45 students play either football, cricket, or both.

Now, the number of students who play neither football nor cricket is:

\[ |U - (F \cup C)| = |U| - |F \cup C| \]

Substituting the values:

\[ |U - (F \cup C)| = 50 - 45 = 5 \]

Therefore, 5 students play neither football nor cricket.