In the group of 240 students, 200 opts for history and 90 opts for Geography. If 20 students do not opt for History as well as Geography, then how many students opt for both the subjects i.e. History and Geography both

Options

A. 70
B. 40
C. 170
D. none of these

Answer

To solve the problem, we will use the principle of inclusion-exclusion along with the information provided about the students opting for History and Geography.

### Step-by-Step Solution:

1. **Identify the total number of students**:
We know that there are 240 students in total.

2. **Identify the number of students opting for each subject**:
- Students opting for History (H) = 200
- Students opting for Geography (G) = 90

3. **Identify the number of students not opting for either subject**:
- Students not opting for History or Geography = 20

4. **Calculate the number of students opting for at least one subject**:
The number of students opting for at least one subject (History or Geography) can be calculated by subtracting the number of students not opting for either subject from the total number of students:
\[ N(H \cup G) = \text{Total Students} - \text{Students not opting for either subject} = 240 - 20 = 220 \]

5. **Use the principle of inclusion-exclusion**:
According to the principle of inclusion-exclusion, the number of students opting for at least one subject can be expressed as:
\[ N(H \cup G) = N(H) + N(G) - N(H \cap G) \]
Where \( N(H \cap G) \) is the number of students opting for both subjects.

6. **Substitute the known values into the equation**:
\[ 220 = 200 + 90 - N(H \cap G) \]

7. **Simplify the equation**:
\[ 220 = 290 - N(H \cap G) \]
Rearranging gives:
\[ N(H \cap G) = 290 - 220 = 70 \]

8. **Conclusion**:
The number of students who opted for both History and Geography is \( N(H \cap G) = 70 \).

### Final Answer:
The number of students who opt for both subjects is **70**.