A survey shows that 63 % of the pepole watch a news channel whereas , 76 % watch an entertiament channel at a particular time If X% of the pepole watch both types of channels , then

Options

A. x= 35
B. `x ge 63`
C. `39 le x le 63`
D. x=39

Explanation

To solve the problem, we need to find the percentage \( X \) of people who watch both the news and entertainment channels based on the given percentages of viewers for each channel.

### Step-by-Step Solution:

1. **Identify the Given Percentages**:
- Let \( N \) be the percentage of people who watch the news channel: \( N = 63\% \).
- Let \( E \) be the percentage of people who watch the entertainment channel: \( E = 76\% \).
- Let \( X \) be the percentage of people who watch both channels.

2. **Use the Principle of Inclusion-Exclusion**:
The formula for the union of two sets is given by:
\[ N \cup E = N + E - N \cap E \]
Here, \( N \cup E \) represents the total percentage of people watching at least one of the channels, and \( N \cap E \) represents the percentage of people watching both channels.

3. **Set Up the Inequality**:
Since the total percentage of viewers cannot exceed 100%, we have:
\[ N + E - X \leq 100 \]
Substituting the known values:
\[ 63 + 76 - X \leq 100 \]

4. **Simplify the Inequality**:
Combine the percentages:
\[ 139 - X \leq 100 \]
Rearranging gives:
\[ X \geq 139 - 100 \]
Thus:
\[ X \geq 39 \]

5. **Determine the Upper Limit for \( X \)**:
The maximum value of \( X \) can be determined by recognizing that \( X \) cannot exceed the smaller of the two percentages \( N \) and \( E \):
\[ X \leq \min(N, E) = \min(63, 76) = 63 \]

6. **Combine the Results**:
From the inequalities, we have:
\[ 39 \leq X \leq 63 \]

### Final Result:
The interval for \( X \) is:
\[ X \in [39, 63] \]