In a class of 55 students the number of students studying subjects are , 23 iin Mathematics 24.in physics 19in chemisty 12 in Mathematics and physics ,9 in Mathematics and chemistry ,7 in physics and chemistry and 4 in all the three subjects.
the number of students who have taken ecactly one subject is

Options

A. 6
B. 9
C. 7
D. none of these

Explanation

To find the number of students who have taken exactly one subject, we can use the principle of inclusion-exclusion. Let's break down the problem step by step.

### Step 1: Define the Variables
Let:
- \( n(M) \) = Number of students studying Mathematics = 23
- \( n(P) \) = Number of students studying Physics = 24
- \( n(C) \) = Number of students studying Chemistry = 19
- \( n(M \cap P) \) = Number of students studying both Mathematics and Physics = 12
- \( n(M \cap C) \) = Number of students studying both Mathematics and Chemistry = 9
- \( n(P \cap C) \) = Number of students studying both Physics and Chemistry = 7
- \( n(M \cap P \cap C) \) = Number of students studying all three subjects = 4

### Step 2: Use the Inclusion-Exclusion Principle
The formula for the number of students who have taken at least one subject is:
\[ n(M \cup P \cup C) = n(M) + n(P) + n(C) - n(M \cap P) - n(M \cap C) - n(P \cap C) + n(M \cap P \cap C) \]

### Step 3: Substitute the Values
Substituting the values we have:
\[ n(M \cup P \cup C) = 23 + 24 + 19 - 12 - 9 - 7 + 4 \]

### Step 4: Calculate the Total
Calculating the above expression step by step:
1. \( 23 + 24 + 19 = 66 \)
2. \( 12 + 9 + 7 = 28 \)
3. \( 66 - 28 + 4 = 42 \)

So, \( n(M \cup P \cup C) = 42 \). This means 42 students are studying at least one subject.

### Step 5: Find the Number of Students Taking Exactly One Subject
To find the number of students taking exactly one subject, we can use the following formula:
\[ n(\text{exactly one subject}) = n(M) + n(P) + n(C) - 2(n(M \cap P) + n(M \cap C) + n(P \cap C)) + 3n(M \cap P \cap C) \]

### Step 6: Substitute the Values
Substituting the values:
\[ n(\text{exactly one subject}) = 23 + 24 + 19 - 2(12 + 9 + 7) + 3(4) \]

### Step 7: Calculate the Exact Number
Calculating step by step:
1. \( 23 + 24 + 19 = 66 \)
2. \( 12 + 9 + 7 = 28 \)
3. \( 2 \times 28 = 56 \)
4. \( 3 \times 4 = 12 \)
5. Now substituting these into the equation:
\[ n(\text{exactly one subject}) = 66 - 56 + 12 = 22 \]

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