Poor Dollys 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.

Explanation

Let there be 4 channels `C_(1),C_(2),C_(3), " and" C_(4)`. At t=0 min, she is watching channel 1
`therefore` After 1st minute, she has 3 choices to switch the channel `(C_(2),C_(3),C_(4))`
After 2nd mintue, she has 2 choices to switch the channel
After 3rd mintue, she has 2 choices to switch the channel
But after the 4th minute, she has only 1 choice to switch the channel, i.e., `C_(1)`
`therefore` Total number of ways`=3xx2xx2=12`