QUESTION
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 that she is back to her original channel for the first time after $4$ minutes.
I am having problem in taking cases to change channels. I am not able to understand how can channel change.
E.g.
Let there be channels $C_1,C_2,C_3,C_4$.
For first minute she will have $3$ choices ($C_2,C_3,C_4$) .Second minute $2$ choices ($C_3,C_4$). Third minute $1$ choices ($C_4$). Fourth minute $1$ choice ($C_1$).
So, according to me its answer must be $6$ ways, but, in my textbook, the answer is $12$ ways.
Where am I wrong?
We illustrate Mike Earnest's comments with a tree diagram.
If we designate the original channel $C_1$, then Dolly has three choices after the first minute, which we call $C_2$, $C_3$, $C_4$. For each such choice, she has two choices after the second minute, namely the two that she has not yet selected. For each such choice, she also has two choices after the third minute, the channel she has not yet selected or the channel she selected after the first minute. Since Dolly must return to her original channel after the fourth minute, she has $3 \cdot 2 \cdot 2 = 12$ ways to return to the original channel for the first time after the four minutes.