Out of 450 students of a school 325 play football, 175 play cricket and 50 neither play football 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 can use the principle of inclusion-exclusion for this.

### 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 the number of students playing each sport**:
- Number of students playing football (F) = 325
- Number of students playing cricket (C) = 175

3. **Identify the number of students playing neither sport**:
The number of students who do not play either sport is given as 50.

4. **Calculate the number of students playing either football or cricket**:
To find the number of students who play at least one of the sports, we subtract the number of students who play neither from the total number of students:
\[ \text{Students playing either football or cricket} = \text{Total students} - \text{Students playing neither} \]
\[ = 450 - 50 = 400 \]

5. **Use the inclusion-exclusion principle**:
According to the principle, the number of students playing either football or cricket can be expressed as:
\[ F + C - \text{Students playing both sports} \]
Let \( x \) be the number of students playing both sports. Therefore, we can write:
\[ 400 = 325 + 175 - x \]

6. **Solve for \( x \)**:
Combine the numbers on the right side:
\[ 400 = 500 - x \]
Rearranging gives:
\[ x = 500 - 400 = 100 \]

7. **Conclusion**:
The number of students who play both football and cricket is \( x = 100 \).

### Final Answer:
100 students play both football and cricket.