In a class of 75 students, the number of students studying different Subjects are 43 in Mathematics 44 in Physics, 39 in Chemistry, 32 in Mathematics and Physics, 29 in Mathematics and Chemistry 27 in Physics and Chemistry and 24 in all three subjects. The number of students who have taken exactly one subject is

Options

A. 26
B. 29
C. 27
D. 22

Answer

To solve the problem of finding the number of students who have taken exactly one subject, we will use the principle of inclusion-exclusion and Venn diagrams.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( n(M) = 43 \) (students studying Mathematics)
- Let \( n(P) = 44 \) (students studying Physics)
- Let \( n(C) = 39 \) (students studying Chemistry)
- Let \( n(M \cap P) = 32 \) (students studying both Mathematics and Physics)
- Let \( n(M \cap C) = 29 \) (students studying both Mathematics and Chemistry)
- Let \( n(P \cap C) = 27 \) (students studying both Physics and Chemistry)
- Let \( n(M \cap P \cap C) = 24 \) (students studying all three subjects)

2. **Calculate the Number of Students in Each Intersection**:
- Students studying only Mathematics and Physics (but not Chemistry):
\[ n(M \cap P) - n(M \cap P \cap C) = 32 - 24 = 8 \]
- Students studying only Mathematics and Chemistry (but not Physics):
\[ n(M \cap C) - n(M \cap P \cap C) = 29 - 24 = 5 \]
- Students studying only Physics and Chemistry (but not Mathematics):
\[ n(P \cap C) - n(M \cap P \cap C) = 27 - 24 = 3 \]

3. **Set Up the Equations for Students Studying Only One Subject**:
- Let \( x \) be the number of students studying only Mathematics.
- Let \( y \) be the number of students studying only Physics.
- Let \( z \) be the number of students studying only Chemistry.

From the total number of students studying Mathematics:
\[ x + (n(M \cap P) - n(M \cap P \cap C)) + (n(M \cap C) - n(M \cap P \cap C)) + n(M \cap P \cap C) = n(M) \]
\[ x + 8 + 5 + 24 = 43 \implies x + 37 = 43 \implies x = 6 \]

From the total number of students studying Physics:
\[ y + (n(M \cap P) - n(M \cap P \cap C)) + (n(P \cap C) - n(M \cap P \cap C)) + n(M \cap P \cap C) = n(P) \]
\[ y + 8 + 3 + 24 = 44 \implies y + 35 = 44 \implies y = 9 \]

From the total number of students studying Chemistry:
\[ z + (n(M \cap C) - n(M \cap P \cap C)) + (n(P \cap C) - n(M \cap P \cap C)) + n(M \cap P \cap C) = n(C) \]
\[ z + 5 + 3 + 24 = 39 \implies z + 32 = 39 \implies z = 7 \]

4. **Calculate the Total Number of Students Studying Exactly One Subject**:
\[ \text{Total} = x + y + z = 6 + 9 + 7 = 22 \]

### Final Answer:
The number of students who have taken exactly one subject is **22**.