A survey was conducted at a coaching institution and it was found that there were 34 students who appeared in MAT. There were 37 students who appeared in CAT of which 17 students appeared in MAT. 30 students appeared in XAT of which 13 students appeared in MAT. Of the XAT applicants (i.e., appeared students) 14 appeared in CAT and these 6 appeared in MAT. How many students appeared in CAT but not in MAT or XAT ?

Options

A. 9
B. 10
C. 12
D. none of these

Answer

To solve the problem, we will use a Venn diagram to visualize the relationships between the students who appeared in the MAT, CAT, and XAT exams.

### Step-by-Step Solution:

1. **Identify the Total Students in Each Exam:**
- Total students who appeared in MAT = 34
- Total students who appeared in CAT = 37
- Total students who appeared in XAT = 30

2. **Identify Overlapping Students:**
- Students who appeared in both MAT and CAT = 17
- Students who appeared in both MAT and XAT = 13
- Students who appeared in both XAT and CAT = 14, out of which 6 also appeared in MAT.

3. **Set Up the Venn Diagram:**
- Let’s denote:
- A = Students who appeared in MAT
- B = Students who appeared in CAT
- C = Students who appeared in XAT
- From the information given:
- |A| = 34
- |B| = 37
- |C| = 30
- |A ∩ B| = 17
- |A ∩ C| = 13
- |B ∩ C| = 14
- |A ∩ B ∩ C| = 6 (students who appeared in all three exams)

4. **Calculate Students in Each Section of the Venn Diagram:**
- For students who appeared only in MAT and CAT:
- |A ∩ B| - |A ∩ B ∩ C| = 17 - 6 = 11
- For students who appeared only in MAT and XAT:
- |A ∩ C| - |A ∩ B ∩ C| = 13 - 6 = 7
- For students who appeared only in CAT and XAT:
- |B ∩ C| - |A ∩ B ∩ C| = 14 - 6 = 8

5. **Fill in the Venn Diagram:**
- Students who appeared only in MAT = |A| - (students in A ∩ B + students in A ∩ C - students in A ∩ B ∩ C)
- = 34 - (11 + 7 + 6) = 10
- Students who appeared only in CAT = |B| - (students in A ∩ B + students in B ∩ C - students in A ∩ B ∩ C)
- = 37 - (11 + 8 + 6) = 12
- Students who appeared only in XAT = |C| - (students in A ∩ C + students in B ∩ C - students in A ∩ B ∩ C)
- = 30 - (7 + 8 + 6) = 9

6. **Calculate Students Appeared in CAT but Not in MAT or XAT:**
- The number of students who appeared in CAT but not in MAT or XAT = Students who appeared only in CAT = 12.

### Final Answer:
The number of students who appeared in CAT but not in MAT or XAT is **12**.