A group of 50 students appeared for the two examinations one in Physics and the other in Mathematics. 38 students passed in Physics and 37 in Mathematics. If 30 students passed in both subjects, determine how many students failed in both the subjects.

Options

A. 2
B. 3
C. 4
D. 5

Answer

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

### Step 1: Identify the total number of students and the number of students passing each subject.
- Total students = 50
- Students passing Physics = 38
- Students passing Mathematics = 37
- Students passing both subjects = 30

### Step 2: Calculate the number of students passing only one subject.
- Students passing only Physics = Students passing Physics - Students passing both
\[ \text{Only Physics} = 38 - 30 = 8 \]
- Students passing only Mathematics = Students passing Mathematics - Students passing both
\[ \text{Only Mathematics} = 37 - 30 = 7 \]

### Step 3: Calculate the total number of students passing at least one subject.
- Total students passing at least one subject = Students passing only Physics + Students passing only Mathematics + Students passing both
\[ \text{Total passing} = 8 + 7 + 30 = 45 \]

### Step 4: Calculate the number of students failing both subjects.
- Students failing both subjects = Total students - Students passing at least one subject
\[ \text{Failing both} = 50 - 45 = 5 \]

### Conclusion:
The number of students who failed in both subjects is **5**.

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