Out of 100 students, 15 passed in English, 12 passed in Mathmatics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science , 4 in English and Science, 4 in all the three. Find how many passed
(i) in English and Mathematics but not in Science.
(ii) in Mathematics and Science but not in English.
(iii) in Mathematics only.
(iv) in more than one subject only.

Explanation

To solve the problem step by step, we will use the principle of inclusion-exclusion and a Venn diagram to visualize the data.

### Given Data:
- Total students = 100
- Students passed in:
- English (E) = 15
- Mathematics (M) = 12
- Science (S) = 8
- English and Mathematics (E ∩ M) = 6
- Mathematics and Science (M ∩ S) = 7
- English and Science (E ∩ S) = 4
- All three subjects (E ∩ M ∩ S) = 4

### Step 1: Fill in the Venn Diagram
1. **All three subjects (E ∩ M ∩ S)**: 4 students passed in all three subjects.
2. **English and Mathematics (E ∩ M)**: 6 students passed in both English and Mathematics. Since 4 of these also passed Science, the number of students who passed only English and Mathematics (E ∩ M but not S) is:
\[ (E \cap M) - (E \cap M \cap S) = 6 - 4 = 2 \]
3. **Mathematics and Science (M ∩ S)**: 7 students passed in both Mathematics and Science. Since 4 of these also passed English, the number of students who passed only Mathematics and Science (M ∩ S but not E) is:
\[ (M \cap S) - (E \cap M \cap S) = 7 - 4 = 3 \]
4. **English and Science (E ∩ S)**: 4 students passed in both English and Science. Since 4 of these also passed Mathematics, the number of students who passed only English and Science (E ∩ S but not M) is:
\[ (E \cap S) - (E \cap M \cap S) = 4 - 4 = 0 \]

### Step 2: Calculate Students Passing Only One Subject
1. **Only English**: Total students who passed English = 15. The students who passed English but not Mathematics or Science:
\[ E - (E \cap M) - (E \cap S) + (E \cap M \cap S) = 15 - 6 - 4 + 4 = 9 \]
2. **Only Mathematics**: Total students who passed Mathematics = 12. The students who passed Mathematics but not English or Science:
\[ M - (E \cap M) - (M \cap S) + (E \cap M \cap S) = 12 - 6 - 7 + 4 = 3 \]
3. **Only Science**: Total students who passed Science = 8. The students who passed Science but not English or Mathematics:
\[ S - (E \cap S) - (M \cap S) + (E \cap M \cap S) = 8 - 4 - 7 + 4 = 1 \]

### Step 3: Summary of Students
- Only English = 9
- Only Mathematics = 3
- Only Science = 1
- English and Mathematics but not Science = 2
- Mathematics and Science but not English = 3
- English and Science but not Mathematics = 0
- All three subjects = 4

### Final Answers
(i) Students who passed in English and Mathematics but not in Science = **2**
(ii) Students who passed in Mathematics and Science but not in English = **3**
(iii) Students who passed in Mathematics only = **3**
(iv) Students who passed in more than one subject only = **2 (E ∩ M) + 3 (M ∩ S) + 4 (E ∩ M ∩ S) = 9**