A class has 172 students. The following data shows the number of students opting one or more subjects: Mathematics 125, Physics 95, Chemistry 65 Mathematics and Physics 55
Mathematics and Chemistry 53
Physics and Chemistry 48 Mathematics, Physics and Chemistry 43
The number of students who opted only Physics is

Options

A. 35
B. 45
C. 29
D. 37

Answer

To find the number of students who opted only for Physics, we can use the principle of inclusion-exclusion along with the given data. Let's break down the solution step by step.

### Step 1: Define the Variables
Let:
- \( n(M) \) = Number of students opting for Mathematics = 125
- \( n(P) \) = Number of students opting for Physics = 95
- \( n(C) \) = Number of students opting for Chemistry = 65
- \( n(M \cap P) \) = Number of students opting for both Mathematics and Physics = 55
- \( n(M \cap C) \) = Number of students opting for both Mathematics and Chemistry = 53
- \( n(P \cap C) \) = Number of students opting for both Physics and Chemistry = 48
- \( n(M \cap P \cap C) \) = Number of students opting for all three subjects = 43

### Step 2: Calculate the Number of Students in Each Intersection
Using the values provided, we can find the number of students in the intersections of the subjects:
1. Students opting for only Mathematics and Physics:
\[ n(M \cap P) - n(M \cap P \cap C) = 55 - 43 = 12 \]

2. Students opting for only Mathematics and Chemistry:
\[ n(M \cap C) - n(M \cap P \cap C) = 53 - 43 = 10 \]

3. Students opting for only Physics and Chemistry:
\[ n(P \cap C) - n(M \cap P \cap C) = 48 - 43 = 5 \]

### Step 3: Set Up the Equation for Students Opting for Only Physics
Let \( x \) be the number of students who opted only for Physics. The total number of students opting for Physics can be expressed as:
\[ x + (n(M \cap P) - n(M \cap P \cap C)) + (n(P \cap C) - n(M \cap P \cap C)) + n(M \cap P \cap C) = n(P) \]
Substituting the known values:
\[ x + 12 + 5 + 43 = 95 \]

### Step 4: Solve for \( x \)
Now, simplify the equation:
\[ x + 60 = 95 \]
Subtract 60 from both sides:
\[ x = 95 - 60 = 35 \]

### Conclusion
The number of students who opted only for Physics is **35**.

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