Q:

probability of getting specific consequence

I roll the dice and puts down the results in his notebook. I want the numbers 3,4,5,2,6,1 to be written in this order. However there can be other numbers in between. How many times on average should I roll the dice in order to get such kind of a result?

A:

Hint
The length of the sequence = $n_1+n_2+n_3+n_4+n_5+n_6+6$, where $n_1$ is the number times that the dice has to be rolled before first 3 occurs, $n_2$  is the number times that the dice has to be rolled before first 4 occurs after a 3, and so on. Clearly, $n_i$ are iid and 
\begin{equation}
P(n_i = n) = \left(\frac{5} {6}\right)^n\frac{1} {6}. 
\end{equation}
After some algebraic simplifications, we get that the expected value is 42.

