If in a class there are 200 students in which 120 take Mathematics, 90 take Physics, 60 take Chemistry, 50 take Mathematics & Physics, 50 take Mathematics & Chemistry, 43 take Physics & Chemistry and 38 take Mathematics Physics & Chemistry, then the number of students who hace taken exactly one subject is

Options

A. 42
B. 56
C. 270
D. 98

Answer

To solve the problem of finding the number of students who have taken exactly one subject among Mathematics, Physics, and Chemistry, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( n(M) = 120 \) (students taking Mathematics)
- \( n(P) = 90 \) (students taking Physics)
- \( n(C) = 60 \) (students taking Chemistry)
- \( n(M \cap P) = 50 \) (students taking both Mathematics and Physics)
- \( n(M \cap C) = 50 \) (students taking both Mathematics and Chemistry)
- \( n(P \cap C) = 43 \) (students taking both Physics and Chemistry)
- \( n(M \cap P \cap C) = 38 \) (students taking all three subjects)

### Step 2: Use the Formula for Exactly One Subject
The formula to find the number of students who have taken exactly one subject is given by:

\[ n(M \text{ only}) + n(P \text{ only}) + n(C \text{ only}) = n(M) + n(P) + n(C) - n(M \cap P) - n(M \cap C) - n(P \cap C) + n(M \cap P \cap C) \]

### Step 3: Calculate Each Component
1. **Calculate students taking only Mathematics**:
\[ n(M \text{ only}) = n(M) - n(M \cap P) - n(M \cap C) + n(M \cap P \cap C) \]
\[ n(M \text{ only}) = 120 - 50 - 50 + 38 = 58 \]

2. **Calculate students taking only Physics**:
\[ n(P \text{ only}) = n(P) - n(M \cap P) - n(P \cap C) + n(M \cap P \cap C) \]
\[ n(P \text{ only}) = 90 - 50 - 43 + 38 = 35 \]

3. **Calculate students taking only Chemistry**:
\[ n(C \text{ only}) = n(C) - n(M \cap C) - n(P \cap C) + n(M \cap P \cap C) \]
\[ n(C \text{ only}) = 60 - 50 - 43 + 38 = 5 \]

### Step 4: Sum the Students Taking Exactly One Subject
Now, we can find the total number of students who have taken exactly one subject:
\[ n(\text{exactly one}) = n(M \text{ only}) + n(P \text{ only}) + n(C \text{ only}) \]
\[ n(\text{exactly one}) = 58 + 35 + 5 = 98 \]

### Final Answer
The number of students who have taken exactly one subject is **98**.