In a class of 175 students the following data shows the number of students opting one or more subjects Mathematics 100, physics 70, chemistry 28, physics and Chemistry 23, mathematics and chemistry 28 , mathematics and physics 30 , mathematics physics and chemistry 18. how many students have offered mathematics alone ?

Options

A. 35
B. 48
C. 60
D. 22

Answer

To find the number of students who have opted for Mathematics alone, we will use the principle of inclusion-exclusion and the given data.

### Step 1: Define the variables
Let:
- \( M \): Number of students who opted for Mathematics = 100
- \( P \): Number of students who opted for Physics = 70
- \( C \): Number of students who opted for Chemistry = 28
- \( x \): Number of students who opted for all three subjects (Mathematics, Physics, Chemistry) = 18
- \( y \): Number of students who opted for both Mathematics and Physics but not Chemistry
- \( z \): Number of students who opted for both Mathematics and Chemistry but not Physics
- \( d \): Number of students who opted for both Physics and Chemistry but not Mathematics

### Step 2: Set up equations based on the given data
From the problem, we have:
1. \( P \cap C = 23 \) (Physics and Chemistry)
- This gives us the equation: \( x + d = 23 \)
- Substituting \( x = 18 \):
\[ 18 + d = 23 \implies d = 5 \]

2. \( M \cap C = 28 \) (Mathematics and Chemistry)
- This gives us the equation: \( x + z = 28 \)
- Substituting \( x = 18 \):
\[ 18 + z = 28 \implies z = 10 \]

3. \( M \cap P = 30 \) (Mathematics and Physics)
- This gives us the equation: \( x + y = 30 \)
- Substituting \( x = 18 \):
\[ 18 + y = 30 \implies y = 12 \]

### Step 3: Calculate the number of students who opted for Mathematics alone
The total number of students who opted for Mathematics can be expressed as:
\[ M = b + y + z + x \]
Where \( b \) is the number of students who opted for Mathematics alone.

Substituting the known values:
\[ 100 = b + 12 + 10 + 18 \]
Simplifying:
\[ 100 = b + 40 \]
Thus,
\[ b = 100 - 40 = 60 \]

### Conclusion
The number of students who have opted for Mathematics alone is **60**.