dolly is watching t.v which has only 4 channels and they are boring. so she changes the channels fter every minute.no.of ways she can change the channels in first 4 minutes so that after 4 minutes she get the oriinal channel back

Dear student

If we designate the original channel C1, then Dolly has three choices after the first minute, which we call C2, C3, C4. 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⋅2⋅2=12 ways to return to the original channel for the first time after the four minutes.

Good Luck