A 5 Dice Game

Expectation is additive, so we can concentrate on one die. Every time we roll a die, the expected outcome is $3.5,$ so the expected sum equals $3.5 \cdot 5 \cdot N_5,$ where $N_5$ is the expected number of times we will roll a given die when it is one of $5.$

We get a system of equations as follows: $\begin{aligned} N_2 &= \frac16 N_2 + 1 \\ N_3 &= \frac1{36} N_3 + \frac{10}{36} N_2 + 1 \\ N_4 &= \frac1{216} N_4 + \frac{15}{216} N_3 + \frac{75}{216} N_2 + 1 \\ N_5 &= \frac1{1296} N_5 + \frac{20}{1296} N_4 + \frac{150}{1296} N_3 + \frac{500}{1296} N_2 + 1 \end{aligned}$ Each of the fractions in the above system of equations is the probability that the die we roll will be part of a group of a given size with the same result. For instance, the coefficient of $N_3$ in the last equation is the probability that the die we roll will be a part of a group of $3$ with the same result, which is the probability that exactly $2$ of the other $4$ dice match our die, which is $\binom{4}{2} \left( \frac16 \right)^2 \left( \frac56 \right)^2 = \frac{150}{1296}.$

The solution of this system is $\begin{aligned} N_2 &= \frac65 \\ N_3 &= \frac{48}{35} \\ N_4 &= \frac{2286}{1505} \\ N_5 &= \frac{18336}{11137} \end{aligned}$ and the expected sum is $\frac{18336}{11137} \cdot \frac{35}2 = \frac{45840}{1591},$ so the answer is $\fbox{47431}.$