Suppose we have a die with $K$ faces with numbers from 1 to $K$ written on it, and integers $L$ and $F$ ($0 < L \leq K$). We roll it $N$ times. Let $a_i$ be the number of times (out of the $N$ rolls) that a face with number $i$ written on it came up as the top face of the die.
I need to find the expectation of the value $a_1^F \times a_2^F \times \cdots a_L^F$
For example, let $N=2, K=6, L=2$ and $F=1$
Then, we roll the $6$-face die $2$ times, and we are interested in the value $a_1 \times a_2$.
The only two possible scenarios when this value is not zero are $(1, 2)$ and $(2, 1)$.
Both of them have $a_1 \times a_2 = 1$ and happen with probability $1 / 36$ each. So $P / Q = (1 + 1) / 36 = 1 / 18$