Poor Dolly’s T.V. has only 4 channels, all of them quite boring. Hence it is not surprising that she desires to switch (change) channel after every one minute. Then find the number of ways in which she can change the channels so the she is back to her original channel for the first time after 4 min.

Options

A. 56
B. 48
C. 64
D. 36

Answer

To solve the problem of how many ways Dolly can change channels such that she is back to her original channel for the first time after 4 minutes, we can break it down into a series of logical steps.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Dolly has 4 channels to choose from.
- She needs to switch channels every minute and return to her original channel after 4 minutes.
- The sequence of channels must start and end with the same channel.

2. **Setting Up the Sequence**:
- Let’s denote the original channel as \( C_1 \).
- The sequence of channels over 4 minutes will look like this: \( C_1 \) (1st minute) → \( C_x \) (2nd minute) → \( C_y \) (3rd minute) → \( C_1 \) (4th minute).
- Here, \( C_x \) and \( C_y \) are the channels that Dolly switches to in the 2nd and 3rd minutes.

3. **Choosing Channels for Middle Positions**:
- The middle channels \( C_x \) and \( C_y \) can be any of the other 3 channels (since she cannot stay on \( C_1 \) for the 2nd and 3rd minutes).
- There are two cases to consider:
- **Case 1**: \( C_x \) and \( C_y \) are different.
- **Case 2**: \( C_x \) and \( C_y \) are the same.

4. **Calculating for Case 1 (Different Channels)**:
- If \( C_x \) and \( C_y \) are different, we can choose \( C_x \) in 3 ways (from the remaining channels) and \( C_y \) in 2 ways (from the remaining channels after choosing \( C_x \)).
- Total ways for this case = \( 3 \times 2 = 6 \).

5. **Calculating for Case 2 (Same Channel)**:
- If \( C_x \) and \( C_y \) are the same, we can choose \( C_x \) (which is the same as \( C_y \)) in 3 ways.
- Total ways for this case = 3.

6. **Total Ways to Fill Middle Channels**:
- Total ways for both cases = Ways from Case 1 + Ways from Case 2 = \( 6 + 3 = 9 \).

7. **Choosing the Starting and Ending Channel**:
- Since the starting and ending channel is \( C_1 \), there are 4 choices for the starting channel (it can be any of the 4 channels).
- Therefore, the number of ways to choose the starting and ending channel is 4.

8. **Final Calculation**:
- The total number of ways Dolly can change channels is given by multiplying the number of ways to fill the middle channels by the number of ways to choose the starting and ending channel.
- Total ways = \( 9 \times 4 = 36 \).

### Final Answer:
The number of ways in which Dolly can change channels so that she is back to her original channel for the first time after 4 minutes is **36**.