Question 148
Nine balls, numbered $1$, $2$, $\ldots$, $9$, are put randomly at $9$ equally spaced points on a circle, each point with a ball. Let $S$ be the sum of the absolute values of the differences of the numbers of all two neighboring balls. Find the probability of $S$ to be the minimum value. (Remark: If one arrangement of the balls is congruent to another after a rotation or a reflection, the two arrangements are regarded as the same).