In a class of `100` students exactly `15` students passed in Physics, exactly `20` stundents passed in Chemistry and exactly `45` students passes in Mathematics. If `x_(1)` and `x_(2)` are of maximum and minimum number of students respectively failed in all 3 subjects then :

Options

A. `x_(1)+x_(2)=75`
B. `x_(1)-x_(2)=45`
C. `x_(1)+x_(2)=85`
D. `x_(1)-x_(2)=35`

Explanation

To solve the problem, we need to determine the maximum and minimum number of students who failed in all three subjects: Physics, Chemistry, and Mathematics.

### Step-by-Step Solution:

1. **Identify the Total Number of Students**:
- Total number of students in the class = 100.

2. **Identify the Number of Students Passing Each Subject**:
- Students passing Physics (P) = 15.
- Students passing Chemistry (C) = 20.
- Students passing Mathematics (M) = 45.

3. **Calculate the Maximum Number of Students Failing All Subjects (x1)**:
- To find the maximum number of students failing all subjects, we consider the scenario where the students passing Physics and Chemistry are also included in those passing Mathematics.
- The maximum number of students who can fail all subjects is calculated as:
\[ x_1 = \text{Total Students} - \text{Students passing Mathematics} = 100 - 45 = 55. \]

4. **Calculate the Minimum Number of Students Failing All Subjects (x2)**:
- To find the minimum number of students failing all subjects, we need to consider the overlap of students passing the subjects.
- We can use the principle of inclusion-exclusion:
\[ x_2 = \text{Total Students} - (\text{Students passing Physics} + \text{Students passing Chemistry} + \text{Students passing Mathematics}) + \text{Students passing all three subjects}. \]
- Since we do not have the number of students passing all three subjects, we can assume the worst-case scenario where there is no overlap. Thus:
\[ x_2 = 100 - (15 + 20 + 45) = 100 - 80 = 20. \]

5. **Conclusion**:
- The maximum number of students failing all subjects (x1) = 55.
- The minimum number of students failing all subjects (x2) = 20.

6. **Final Calculation**:
- The difference between the maximum and minimum number of students failing all subjects is:
\[ x_1 - x_2 = 55 - 20 = 35. \]

### Summary:
- Maximum number of students failing all subjects (x1) = 55.
- Minimum number of students failing all subjects (x2) = 20.
- The difference \( x_1 - x_2 = 35 \).