A class has `175` students. The following data shows the number of students obtaining one or more subjects : Mathematics `100`, Physics `70`, Chemistry `40`, Mathematics and Physics `30`, Mathematics and Chemistry `28`, Physics and Chemistry `23`, Mathematics & Physics & Chemistry `18`. How many students have offered Mathematics alone ?

Answer

To find the number of students who have opted for Mathematics alone, we can use the principle of inclusion-exclusion. Let's denote:

- \( M \): the set of students who study Mathematics
- \( P \): the set of students who study Physics
- \( C \): the set of students who study Chemistry

From the problem, we have the following data:

- \( |M| = 100 \) (students studying Mathematics)
- \( |P| = 70 \) (students studying Physics)
- \( |C| = 40 \) (students studying Chemistry)
- \( |M \cap P| = 30 \) (students studying both Mathematics and Physics)
- \( |M \cap C| = 28 \) (students studying both Mathematics and Chemistry)
- \( |P \cap C| = 23 \) (students studying both Physics and Chemistry)
- \( |M \cap P \cap C| = 18 \) (students studying all three subjects)

We want to find the number of students who study only Mathematics, denoted as \( |M \text{ only}| \).

### Step 1: Use the formula for students studying only Mathematics
The formula for the number of students studying only Mathematics is given by:

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

### Step 2: Substitute the values into the formula
Substituting the values we have:

\[ |M \text{ only}| = 100 - (30 + 28 - 18) \]

### Step 3: Simplify the expression
Now, simplify the expression inside the parentheses:

\[ |M \text{ only}| = 100 - (30 + 28 - 18) = 100 - (40) = 100 - 40 \]

### Step 4: Calculate the final result
Now, we can calculate the final result:

\[ |M \text{ only}| = 100 - 40 = 60 \]

Thus, the number of students who have opted for Mathematics alone is **60**.

---