A group of people were surveyed one of the choose one of the tow TV shows they preferred more : Sherlock Holmes and Friends . Of the total 30 people who put forward their choice , 18 chose Sherlck Holmes and 20 chose Friends.Each preson had to choose one of the shows mentioned above . Choose the correct statements (s) :
I. 8 people liked both shows.
II. 22 people perferred one show over the other .
III. 12 people liked only one of the two shows.

Options

A. Only I
B. Only II
C. Only III
D. Both I and II

Answer

To solve the problem, we will analyze the information given step by step:

1. **Understanding the Total Surveyed**:
- Total number of people surveyed = 30
- Number of people who chose Sherlock Holmes = 18
- Number of people who chose Friends = 20

2. **Calculating the Overlap**:
- If we add the number of people who chose both shows, we get 18 + 20 = 38. However, since only 30 people were surveyed, this means that some people must have chosen both shows.
- Let \( x \) be the number of people who liked both shows. Therefore, we can set up the equation:
\[ 18 + 20 - x = 30 \]
- Simplifying this gives:
\[ 38 - x = 30 \]
\[ x = 38 - 30 = 8 \]
- Thus, 8 people liked both shows.

3. **Evaluating the Statements**:
- **Statement I**: "8 people liked both shows."
- This is true based on our calculation.
- **Statement II**: "22 people preferred one show over the other."
- We can find this by subtracting the number of people who liked both shows from the total:
\[ 30 - 8 = 22 \]
- This statement is also true.
- **Statement III**: "12 people liked only one of the two shows."
- To find the number of people who liked only one show, we can calculate:
- People who liked only Sherlock Holmes = 18 - 8 = 10
- People who liked only Friends = 20 - 8 = 12
- Total people who liked only one show = 10 + 12 = 22
- This statement is false because it contradicts our previous finding.

4. **Conclusion**:
- The correct statements are I and II. Therefore, the answer is that statements I and II are correct.

### Summary of the Correct Statements:
- I. 8 people liked both shows. (True)
- II. 22 people preferred one show over the other. (True)
- III. 12 people liked only one of the two shows. (False)

### Final Answer:
The correct options are I and II.

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