$3$ indistinguishable dice are rolled twice. What is the probability that both throws have the same configuration? > $3$ indistinguishable dice are rolled twice, independently. If each face of each die has the equal probability to appear uppermost, what is the probability that both the throws have the same configuration? What is the sample space of this experiment? For distinguishable I know that the sample space has size $6^6$. What will happen for indistinguishable case? Please help me in this regard. Thank you very much.

The first throw has $6^3$ outcomes (some are equivalent as to configuration).

There are $\binom{6}{1}=6$ outcomes where all dices are the same, so in the next throw chances are only $1$ in $6^3$ you'll get the same configuration.

There are $\binom{6}{3}3!=120$ outcomes where all dices are distinct, and the next throw has $3!$ outcomes with the same configuration (differently permuted which is ignored/indistinguishable).

The remaining $6^3-6-120=90$ outcomes have the form $(a,a,b)$ or $(a,b,b)$. In each such case the next throw has $3!/2!=3$ possible outcomes with the same configuration.

The weighted average of the probabilities in each case is

$$\frac{6\dfrac{1}{6^3}+120\dfrac{3!}{6^3}+90\dfrac{3}{6^3}}{6^3}=\frac{83}{3888}\approx 2.135\%$$