In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. Find the number of students who have taken exactly one subject.

Options

A. 6
B. 7
C. 9
D. 22

Answer

To solve the problem of finding the number of students who have taken exactly one subject in a class of 55 students, we can use the principle of inclusion-exclusion. Let's break down the solution 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: Calculate the Number of Students Studying Only Mathematics
To find the number of students studying only Mathematics, we can use the formula:
\[ n(M \text{ only}) = n(M) - n(M \cap P) - n(M \cap C) + n(M \cap P \cap C) \]
Substituting the values:
\[ n(M \text{ only}) = 23 - 12 - 9 + 4 = 6 \]

### Step 3: Calculate the Number of Students Studying Only Chemistry
Using a similar approach, we find the number of students studying only Chemistry:
\[ n(C \text{ only}) = n(C) - n(M \cap C) - n(P \cap C) + n(M \cap P \cap C) \]
Substituting the values:
\[ n(C \text{ only}) = 19 - 9 - 7 + 4 = 7 \]

### Step 4: Calculate the Number of Students Studying Only Physics
Now, we calculate the number of students studying only Physics:
\[ n(P \text{ only}) = n(P) - n(M \cap P) - n(P \cap C) + n(M \cap P \cap C) \]
Substituting the values:
\[ n(P \text{ only}) = 24 - 12 - 7 + 4 = 9 \]

### Step 5: Calculate the Total Number of Students Studying Exactly One Subject
Now, we sum the number of students studying only Mathematics, only Chemistry, and only Physics:
\[ n(\text{exactly one subject}) = n(M \text{ only}) + n(C \text{ only}) + n(P \text{ only}) \]
Substituting the values:
\[ n(\text{exactly one subject}) = 6 + 7 + 9 = 22 \]

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

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