In an examination out of 100 students, 75 passed in English 60 passed in Mathematics and 45 passed in both English and Mathematics. What is the number of students passed in exactly one of the two subjects?

Options

A. 45
B. 60
C. 75
D. 90

Answer

To solve the problem, we can use the principle of set theory and Venn diagrams. Let's denote:

- Let \( E \) be the set of students who passed in English.
- Let \( M \) be the set of students who passed in Mathematics.

From the information given:
- Total number of students, \( n = 100 \)
- Number of students who passed in English, \( |E| = 75 \)
- Number of students who passed in Mathematics, \( |M| = 60 \)
- Number of students who passed in both subjects, \( |E \cap M| = 45 \)

We need to find the number of students who passed in exactly one of the two subjects. This can be calculated using the formula:

\[ \text{Number of students passed in exactly one subject} = |E| + |M| - 2|E \cap M| \]

### Step-by-step Solution:

1. **Calculate the number of students who passed only in English**:
\[ |E \text{ only}| = |E| - |E \cap M| = 75 - 45 = 30 \]

2. **Calculate the number of students who passed only in Mathematics**:
\[ |M \text{ only}| = |M| - |E \cap M| = 60 - 45 = 15 \]

3. **Add the number of students who passed only in English and only in Mathematics**:
\[ \text{Number of students passed in exactly one subject} = |E \text{ only}| + |M \text{ only}| = 30 + 15 = 45 \]

Thus, the number of students who passed in exactly one of the two subjects is **45**.