Read the data given below and answer the quesitons based on it.
There were 3 sections namely A, B and C in a test. Out of three sections, 33 passed section A, 34 students passed in Section B and 32 passed in Section C. 10 students passed in Section A and Section B, 9 passed in Section B and Section C, 8 passed in Section A and Section C. The number of students who passed each seciton alone was equal and was 21 for each section.
How many passed all the three sections ?
A. 3
B. 6
C. 5
D. 7

Options

A. B
B. A
C. C
D. D

Answer

To solve the problem step by step, we will use a Venn diagram approach to visualize the number of students passing each section and the overlaps between them.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( x \) be the number of students who passed all three sections (A, B, and C).
- Let’s denote:
- \( A \): Students who passed section A
- \( B \): Students who passed section B
- \( C \): Students who passed section C

2. **Given Data**:
- Total students passed in section A = 33
- Total students passed in section B = 34
- Total students passed in section C = 32
- Students who passed both A and B = 10
- Students who passed both B and C = 9
- Students who passed both A and C = 8
- Students who passed only section A = 21
- Students who passed only section B = 21
- Students who passed only section C = 21

3. **Set Up the Venn Diagram**:
- For section A:
- Students who passed only A = 21
- Students who passed A and B but not C = \( 10 - x \)
- Students who passed A and C but not B = \( 8 - x \)
- Students who passed all three sections = \( x \)

- For section B:
- Students who passed only B = 21
- Students who passed A and B but not C = \( 10 - x \)
- Students who passed B and C but not A = \( 9 - x \)
- Students who passed all three sections = \( x \)

- For section C:
- Students who passed only C = 21
- Students who passed A and C but not B = \( 8 - x \)
- Students who passed B and C but not A = \( 9 - x \)
- Students who passed all three sections = \( x \)

4. **Equation for Section A**:
- The total number of students who passed section A can be expressed as:
\[ 21 + (10 - x) + (8 - x) + x = 33 \]
Simplifying this:
\[ 21 + 10 - x + 8 - x + x = 33 \]
\[ 39 - x = 33 \]
\[ x = 39 - 33 = 6 \]

5. **Conclusion**:
- The number of students who passed all three sections is \( x = 6 \).

### Final Answer:
The number of students who passed all three sections is **6**.