In a class of 30 students 12 take needle work, 16 take physics and 18 take history. If all the 30 students take at least one subject and no one takes all three, then the number of students taking 2 subjects is

Options

A. 16
B. 6
C. 8
D. 20

Explanation

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

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( N \) be the number of students taking Needlework.
- Let \( P \) be the number of students taking Physics.
- Let \( H \) be the number of students taking History.
- From the problem, we have:
- \( N = 12 \)
- \( P = 16 \)
- \( H = 18 \)

2. **Total Students**:
- The total number of students in the class is given as 30.

3. **Inclusion-Exclusion Principle**:
- According to the principle of inclusion-exclusion for three sets, we have:
\[ N \cup P \cup H = N + P + H - (N \cap P) - (P \cap H) - (N \cap H) + (N \cap P \cap H) \]
- Since no student takes all three subjects, \( N \cap P \cap H = 0 \).

4. **Set Up the Equation**:
- Substitute the known values into the equation:
\[ 30 = 12 + 16 + 18 - (N \cap P) - (P \cap H) - (N \cap H) + 0 \]
- Simplifying this gives:
\[ 30 = 46 - (N \cap P) - (P \cap H) - (N \cap H) \]

5. **Rearranging the Equation**:
- Rearranging the equation to isolate the intersections:
\[ (N \cap P) + (P \cap H) + (N \cap H) = 46 - 30 \]
- Therefore:
\[ (N \cap P) + (P \cap H) + (N \cap H) = 16 \]

6. **Conclusion**:
- The total number of students taking exactly two subjects is \( (N \cap P) + (P \cap H) + (N \cap H) = 16 \).

### Final Answer:
The number of students taking 2 subjects is **16**.