Out of 450 students of a school 325 play football, 175 play cricket and 50 neither play foot ball nor cricket. How many students play both football and cricket ?

Options

A. 50
B. 100
C. 75
D. 225

Answer

To solve the problem, we need to determine how many students play both football and cricket. We will use the principle of inclusion-exclusion.

### Step-by-Step Solution:

1. **Identify the Total Number of Students**:
The total number of students in the school is given as 450.

2. **Identify Students Playing Football and Cricket**:
- Students playing football = 325
- Students playing cricket = 175
- Students playing neither sport = 50

3. **Calculate Students Playing at Least One Sport**:
To find the number of students who play at least one sport (either football or cricket), we subtract the number of students who play neither sport from the total number of students:
\[ \text{Students playing at least one sport} = \text{Total Students} - \text{Students playing neither} \]
\[ = 450 - 50 = 400 \]

4. **Set Up the Equation Using Inclusion-Exclusion Principle**:
Let \( x \) be the number of students who play both football and cricket. According to the inclusion-exclusion principle:
\[ \text{Students playing at least one sport} = \text{Students playing football} + \text{Students playing cricket} - \text{Students playing both} \]
Substituting the known values:
\[ 400 = 325 + 175 - x \]

5. **Simplify the Equation**:
Combine the numbers on the right side:
\[ 400 = 500 - x \]

6. **Solve for \( x \)**:
Rearranging the equation gives:
\[ x = 500 - 400 \]
\[ x = 100 \]

7. **Conclusion**:
Therefore, the number of students who play both football and cricket is \( \boxed{100} \).