Out of the 60 students of class ,29 took Mathematics ,32 took Biology and 8 took none of the two subjects . How many students took both Mathermatics and Biology ?

Options

A. 8
B. 9
C. 7
D. 6

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion.

### Step-by-Step Solution:

1. **Identify the total number of students**:
- Total students = 60

2. **Identify the number of students taking each subject**:
- Students taking Mathematics (M) = 29
- Students taking Biology (B) = 32
- Students taking neither subject = 8

3. **Calculate the number of students taking at least one subject**:
- Students taking at least one subject = Total students - Students taking neither
- Students taking at least one subject = 60 - 8 = 52

4. **Use the principle of inclusion-exclusion**:
- According to the principle of inclusion-exclusion, the number of students taking at least one subject can be expressed as:
\[ N(M \cup B) = N(M) + N(B) - N(M \cap B) \]
- Where \(N(M \cup B)\) is the number of students taking at least one subject, \(N(M)\) is the number of students taking Mathematics, \(N(B)\) is the number of students taking Biology, and \(N(M \cap B)\) is the number of students taking both subjects.

5. **Substitute the known values into the equation**:
- We have:
\[ 52 = 29 + 32 - N(M \cap B) \]

6. **Simplify the equation**:
- Combine the numbers:
\[ 52 = 61 - N(M \cap B) \]

7. **Solve for \(N(M \cap B)\)**:
- Rearranging gives:
\[ N(M \cap B) = 61 - 52 \]
- Therefore:
\[ N(M \cap B) = 9 \]

8. **Conclusion**:
- The number of students who took both Mathematics and Biology is **9**.