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 .
If no students failed in all three subject , what is the total number of students who passed in QA only , DI only and English only ?

Options

A. 50
B. 30
C. 25
D. can't be determined

Answer

To solve the problem step by step, we can use the principle of inclusion-exclusion and a Venn diagram to visualize the data. Here’s how we can break it down:

### Step 1: Define the Sets
Let:
- A = Students who passed in Quantitative Aptitude (QA)
- B = Students who passed in Data Interpretation (DI)
- C = Students who passed in English

### Step 2: Write Down the Given Information
From the problem, we have:
- |A ∩ B| = 25 (students passed in QA and DI)
- |B ∩ C| = 25 (students passed in DI and English)
- |A ∩ C| = 20 (students passed in QA and English)
- |A ∩ B ∩ C| = 10 (students passed in all three subjects)
- Total students = 80

### Step 3: Calculate Students Passing in Exactly Two Subjects
We need to find the number of students who passed in exactly two subjects:
- Students who passed in QA and English only = |A ∩ C| - |A ∩ B ∩ C| = 20 - 10 = 10
- Students who passed in DI and English only = |B ∩ C| - |A ∩ B ∩ C| = 25 - 10 = 15
- Students who passed in QA and DI only = |A ∩ B| - |A ∩ B ∩ C| = 25 - 10 = 15

### Step 4: Calculate the Total Students Passing in At Least Two Subjects
Now, we sum the students who passed in exactly two subjects and those who passed in all three:
- Total passing in at least two subjects = (QA and DI only) + (DI and English only) + (QA and English only) + (All three)
- Total = 15 + 15 + 10 + 10 = 50

### Step 5: Calculate Students Passing in Only One Subject
To find the number of students who passed in only one subject, we can use the total number of students:
- Students passing in only one subject = Total students - Students passing in at least two subjects
- Students passing in only one subject = 80 - 50 = 30

### Final Answer
Thus, the total number of students who passed in QA only, DI only, and English only is **30**.

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