In a class of 45 students each one plays either Cricket or Hockey. If 30 play Cricket and 27 play Hockey then :
how many play both Cricket and Hockey ?

Answer

To solve the problem, we will use the principle of sets. We need to find out how many students play both Cricket and Hockey.

### Step-by-Step Solution:

1. **Identify the Sets**:
- Let \( A \) be the set of students who play Cricket.
- Let \( B \) be the set of students who play Hockey.

2. **Given Information**:
- Total number of students, \( |A \cup B| = 45 \)
- Number of students who play Cricket, \( |A| = 30 \)
- Number of students who play Hockey, \( |B| = 27 \)

3. **Use the Formula for Union of Sets**:
The formula for the union of two sets is:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
where \( |A \cap B| \) is the number of students who play both Cricket and Hockey.

4. **Substitute the Known Values**:
Substitute the values we have into the formula:
\[ 45 = 30 + 27 - |A \cap B| \]

5. **Solve for \( |A \cap B| \)**:
Rearranging the equation to find \( |A \cap B| \):
\[ |A \cap B| = 30 + 27 - 45 \]
\[ |A \cap B| = 57 - 45 \]
\[ |A \cap B| = 12 \]

6. **Conclusion**:
Therefore, the number of students who play both Cricket and Hockey is \( 12 \).