In an examination `40%` of the students failed in physics, `45%` of the students failed in chemistry `60%` of the students failed in biology. If `20%` of the students failed in physics and chemistry, `25%` of the students failed in chemistry and biology and `8%` of the students failed in all three subjects. Find the `%` of students who passed in all three subjects ?

Options

A. `7%`
B. `8%`
C. `6%`
D. `9%`

Answer

To solve the problem, we will use the principle of inclusion-exclusion to find the percentage of students who passed in all three subjects.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( P \) be the percentage of students who failed in Physics = 40%
- Let \( C \) be the percentage of students who failed in Chemistry = 45%
- Let \( B \) be the percentage of students who failed in Biology = 60%
- Let \( PC \) be the percentage of students who failed in both Physics and Chemistry = 20%
- Let \( CB \) be the percentage of students who failed in both Chemistry and Biology = 25%
- Let \( PB \) be the percentage of students who failed in both Physics and Biology (unknown for now)
- Let \( PCB \) be the percentage of students who failed in all three subjects = 8%

2. **Use Inclusion-Exclusion Principle**:
The formula to find the total percentage of students who failed in at least one subject is:
\[ P + C + B - PC - CB - PB + PCB \]
We need to find \( PB \) (the percentage of students who failed in both Physics and Biology).

3. **Substituting Known Values**:
We know:
- \( P = 40\% \)
- \( C = 45\% \)
- \( B = 60\% \)
- \( PC = 20\% \)
- \( CB = 25\% \)
- \( PCB = 8\% \)

Plugging these values into the inclusion-exclusion formula gives:
\[ 40 + 45 + 60 - 20 - 25 - PB + 8 \]

4. **Simplifying the Equation**:
Combine the known percentages:
\[ 40 + 45 + 60 + 8 - 20 - 25 - PB = 108 - PB \]

5. **Setting Up the Equation**:
Since the total percentage of students cannot exceed 100%, we set up the equation:
\[ 108 - PB \leq 100 \]

6. **Solving for PB**:
Rearranging gives:
\[ PB \geq 8 \]
This means at least 8% of students failed in both Physics and Biology.

7. **Finding the Percentage of Students Who Passed All Subjects**:
To find the percentage of students who passed all subjects, we need to calculate:
\[ \text{Percentage who passed} = 100\% - \text{Percentage who failed in at least one subject} \]
Using \( PB = 8\% \) (the minimum value):
\[ \text{Percentage who failed} = 108 - 8 = 100\% \]

Thus, the percentage of students who passed in all three subjects is:
\[ 100\% - 100\% = 0\% \]

### Conclusion:
The percentage of students who passed in all three subjects is **0%**.