Answer

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**Hint:**In the solution of this question, we are going to use the techniques of permutation and combination for solving. First, we will find the choices of switching channels from the first channel to the next three channels. After that, we will find the choices of switching channels from any of those three channels to any two channels. From here, we will find the choices for switching the rest channels and get the answer.

**Complete step by step answer:**

Let us solve this question.

In this question it is given that the T.V. of Dolly has only 4 channels. And, she changes the channels after every one minute. We have to find the number of ways in which she can change the channels so that she is back to her original or initial channel for the first time. That means she is changing the 4 channels after every one minute and coming back to the initial channel for the first time. So, we have to find the number of ways of changing the channels and coming back to the channel from where she started switching.

So, let us say that the channels A, B, C, and D.

The first channel is A and the last channel is D.

When she starts switching from channel A, she has 3 choices (that are channels B, C, and D) for switching. Before switching from A, she has 4 minutes. Now, she has 3 minutes after switching from A.

Here, we get the choices as 3.

Now, suppose she has reached a channel (any of B, C, and D), and now she is going to switch from those 3 channels to the rest 2 channels, then she will have only 2 choices. (Suppose, she has reached the channel B, then she will have only channels C and D for switching, so she will only have two choices)

We cannot choose channel A, we have to reach channel A after 4 minutes for the first time.

From here, we get 2 choices.

Now, she has 2 minutes to switch.

Now, she has switched any of 2 channels. Suppose, she has chosen C from the channels C and D. Then, for switching the next channel, she will have only 2 choices. (Suppose, she is in channel B, then she has 2 channels that are C and D to switch.)

From here, we get 2 choices.

Now, she has only one minute for switching. In the last minute, she will switch to A and it will have only one choice.

From here, we get only one choice.

Therefore, total choices or ways in which she will change the channels so that she is back to her original channel for the first time after 4 minutes will be

\[3\times 2\times 2\times 1=12\]

**Note:**As we can see that this question is from the topic of permutation and combination, so we should have a better knowledge in that topic. Don’t forget that, she cannot go to the first channel before completing the 4 minutes.

We can understand the solution from a different method.

Drawing the tree diagram for switching the channels, we get

Methods of solving are the same but we have used here a tree diagram.

From here, we can see that there are 12 ways.

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