In a class of 80 students 25 passed in QA and DI , 25 passed in DI and English , 20 passed in QA and English 10 student passed in all the there subjects .
How many students passed only in Q.A ?

Options

A. A. 20
B. B. 15
C. C.12
D. D. can't be determined

Answer

To solve the problem step by step, we will use the information provided about the students passing in different subjects and apply the principles of set theory, particularly focusing on the Venn diagram approach.

### Step-by-Step Solution:

1. **Identify the sets and intersections:**
- Let \( A \) be the set of students who passed in QA.
- Let \( B \) be the set of students who passed in DI.
- Let \( C \) be the set of students who passed in English.
- We know:
- \( |A \cap B| = 25 \) (students passed in QA and DI)
- \( |B \cap C| = 25 \) (students passed in DI and English)
- \( |A \cap C| = 20 \) (students passed in QA and English)
- \( |A \cap B \cap C| = 10 \) (students passed in all three subjects)

2. **Calculate the number of students who passed in exactly two subjects:**
- For QA and DI (but not English):
\[ |A \cap B| - |A \cap B \cap C| = 25 - 10 = 15 \]
- For DI and English (but not QA):
\[ |B \cap C| - |A \cap B \cap C| = 25 - 10 = 15 \]
- For QA and English (but not DI):
\[ |A \cap C| - |A \cap B \cap C| = 20 - 10 = 10 \]

3. **Set up the Venn diagram:**
- Let:
- \( x \) = number of students who passed only in QA
- \( y \) = number of students who passed only in DI
- \( z \) = number of students who passed only in English
- The Venn diagram sections can be represented as:
- Only QA: \( x \)
- Only DI: \( y \)
- Only English: \( z \)
- QA and DI only: \( 15 \)
- DI and English only: \( 15 \)
- QA and English only: \( 10 \)
- All three subjects: \( 10 \)

4. **Write the equation for the total number of students:**
- The total number of students is 80, so we can write:
\[ x + y + z + 15 + 15 + 10 + 10 = 80 \]
- Simplifying this gives:
\[ x + y + z + 60 = 80 \]
- Therefore:
\[ x + y + z = 20 \]

5. **Express the number of students passing only in QA:**
- We need to find \( x \), the number of students who passed only in QA. However, we need more information about \( y \) and \( z \) to find \( x \).

6. **Conclusion:**
- Since we do not have enough information to determine the exact values of \( y \) and \( z \), we cannot find the exact number of students who passed only in QA.

### Final Answer:
We cannot determine the exact number of students who passed only in QA with the information given.