I am copying one example problem from Sheldon Ross which I am not understanding :
Suppose we are given a set of $n$ elements, numbered $1$ through $n$, which are to be arranged in some ordered list. At each unit of time a request is made to retrieve one of these elements, element $i$ being requested (independently of the past) with probability $P_i$. After being requested, the element then is put back one closer to the front of the list.
We can describe the system a Markov Chain with $n!$ states with the state at any time being the list order at that time. Now it is claimed that for the path from a state to itself the product of the transition probabilities for the path is $$\prod_i P_i^{\,f_i + r_i}$$ where $f_i$ denotes the number of times element $i$ moves forward in the path and $r_i$ is equal to the number of times that element $i $ is in the first position and the path does not change states.
I could not understand why $r_i$ is needed ?. I thought it should be $$\prod_i P_i^{\, f_i}$$.