Read the following information carefully and answer the questions given below-
1000 students appeared in an examination Out of which 275 students passed in physics, 320 students passed in chemistry and 350 students passed in mathematics. 84 students passed in both chemistry and mathematics 60 students passed in both physics and mathematics, 50 students passed in both physics and chemistry and 34 students passed in all the three subjects.
What is the percentage of students who did not pass ?

Options

A. 28.3
B. 21.5
C. 28.28
D. 31.7

Answer

To solve the problem, we will use the principle of inclusion-exclusion to find the total number of students who passed at least one subject. Then, we will calculate the percentage of students who did not pass any subject.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Total students (N) = 1000
- Students passed in Physics (P) = 275
- Students passed in Chemistry (C) = 320
- Students passed in Mathematics (M) = 350
- Students passed in both Chemistry and Mathematics (C ∩ M) = 84
- Students passed in both Physics and Mathematics (P ∩ M) = 60
- Students passed in both Physics and Chemistry (P ∩ C) = 50
- Students passed in all three subjects (P ∩ C ∩ M) = 34

2. **Apply the Inclusion-Exclusion Principle:**
The formula for the number of students who passed at least one subject is:
\[ N(P \cup C \cup M) = N(P) + N(C) + N(M) - N(P \cap C) - N(C \cap M) - N(P \cap M) + N(P \cap C \cap M) \]

Plugging in the values:
\[ N(P \cup C \cup M) = 275 + 320 + 350 - 50 - 84 - 60 + 34 \]

3. **Calculate the Total:**
- First, sum the individual subjects:
\[ 275 + 320 + 350 = 945 \]
- Then, subtract the overlaps:
\[ 945 - (50 + 84 + 60) + 34 = 945 - 194 + 34 \]
- Calculate:
\[ 945 - 194 = 751 \]
\[ 751 + 34 = 785 \]

Therefore, the total number of students who passed at least one subject is **785**.

4. **Calculate the Number of Students Who Did Not Pass:**
To find the number of students who did not pass any subject:
\[ \text{Students who did not pass} = N - N(P \cup C \cup M) \]
\[ = 1000 - 785 = 215 \]

5. **Calculate the Percentage of Students Who Did Not Pass:**
The percentage of students who did not pass is given by:
\[ \text{Percentage} = \left( \frac{\text{Number of students who did not pass}}{N} \right) \times 100 \]
\[ = \left( \frac{215}{1000} \right) \times 100 = 21.5\% \]

### Final Answer:
The percentage of students who did not pass is **21.5%**.