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. 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.

• A. $6$
• B. $12$
• C. $3$
• D. $24$

Answer 1

The correct option is B $12$

Let there be $4$ channels $C_1,C_2,C_3$ and $C_4.$

At $t=0$ minute assume she is watching channel $C_1.$

After first minute she has $3$ choices to switch the channel i.e., $C_2,C_3,C_4$

After second and third minute she has $2$ choices to switch the channel since she can't watch $C_1$ and channel which she is already watching before changing.
Now,after the fourth minute she has only $1$ choice to switch the channel i.e., $C_1$

$\therefore$ Total number of ways for switching channels is $=3\times 2\times 2\times 1=12$ ways