In a class of 50 students, 35 opted for Mathematics and 37 opted for Biology . How have opted for only Mathematics ? (Assume that each student has to opt for at least one of the subjects )

Options

A. 15
B. 17
C. 13
D. 19

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the problem
We have a total of 50 students in a class. Out of these, 35 students opted for Mathematics and 37 students opted for Biology. We need to find out how many students opted for only Mathematics.

### Step 2: Use the formula for the union of sets
We know that the total number of students who opted for at least one subject (Mathematics or Biology) is given by the formula:
\[ n(M \cup B) = n(M) + n(B) - n(M \cap B) \]
Where:
- \( n(M \cup B) \) is the total number of students who opted for at least one subject (50).
- \( n(M) \) is the number of students who opted for Mathematics (35).
- \( n(B) \) is the number of students who opted for Biology (37).
- \( n(M \cap B) \) is the number of students who opted for both subjects.

### Step 3: Substitute the known values into the formula
Now we can substitute the values into the formula:
\[ 50 = 35 + 37 - n(M \cap B) \]

### Step 4: Solve for \( n(M \cap B) \)
Rearranging the equation gives us:
\[ n(M \cap B) = 35 + 37 - 50 \]
\[ n(M \cap B) = 72 - 50 \]
\[ n(M \cap B) = 22 \]

### Step 5: Find the number of students who opted for only Mathematics
To find the number of students who opted for only Mathematics, we use the formula:
\[ n(\text{Only } M) = n(M) - n(M \cap B) \]
Substituting the known values:
\[ n(\text{Only } M) = 35 - 22 \]
\[ n(\text{Only } M) = 13 \]

### Conclusion
Thus, the number of students who opted for only Mathematics is **13**.