In the class of 100 students, the number of students passed in English is 46, in Maths is 46 and in Commerce is 58. The number who passed in English and Maths is 16, Maths and commerce is 24 and English and commerce is 26, and the number who passed in all subjects is 7. Find the number of the students who failed in all subjects.

Options

A. 13
B. 8
C. 9
D. 7

Answer

To solve the problem, we will use the principle of inclusion-exclusion and a Venn diagram approach. Here are the steps to find the number of students who failed in all subjects:

### Step 1: Define the Variables
Let:
- \( E \) = number of students who passed in English = 46
- \( M \) = number of students who passed in Maths = 46
- \( C \) = number of students who passed in Commerce = 58
- \( EM \) = number of students who passed in both English and Maths = 16
- \( MC \) = number of students who passed in both Maths and Commerce = 24
- \( EC \) = number of students who passed in both English and Commerce = 26
- \( EMC \) = number of students who passed in all three subjects = 7
- \( N \) = total number of students = 100

### Step 2: Calculate Students Passing Only Two Subjects
Using the values provided, we can find the number of students who passed exactly two subjects.

1. **Students who passed only English and Maths**:
\[ EM - EMC = 16 - 7 = 9 \]

2. **Students who passed only Maths and Commerce**:
\[ MC - EMC = 24 - 7 = 17 \]

3. **Students who passed only English and Commerce**:
\[ EC - EMC = 26 - 7 = 19 \]

### Step 3: Calculate Students Passing Only One Subject
Now, we can find the number of students who passed only one subject.

1. **Students who passed only English**:
\[ E - (EM + EC - EMC) = 46 - (16 + 26 - 7) = 46 - 35 = 11 \]

2. **Students who passed only Maths**:
\[ M - (EM + MC - EMC) = 46 - (16 + 24 - 7) = 46 - 33 = 13 \]

3. **Students who passed only Commerce**:
\[ C - (EC + MC - EMC) = 58 - (26 + 24 - 7) = 58 - 43 = 15 \]

### Step 4: Total Students Passing at Least One Subject
Now, we can sum all the students who passed at least one subject:
- Students who passed only English: 11
- Students who passed only Maths: 13
- Students who passed only Commerce: 15
- Students who passed only English and Maths: 9
- Students who passed only Maths and Commerce: 17
- Students who passed only English and Commerce: 19
- Students who passed all three subjects: 7

Total passing students:
\[ 11 + 13 + 15 + 9 + 17 + 19 + 7 = 91 \]

### Step 5: Calculate Students Who Failed All Subjects
Finally, to find the number of students who failed in all subjects, we subtract the total number of students who passed from the total number of students:
\[ N - \text{Total passing students} = 100 - 91 = 9 \]

### Conclusion
The number of students who failed in all subjects is **9**.