Along the circle there are two routes from 1 to 9, the major arc and the minimum arc for each of them

Let $x_1,x_2,\mathrm{......}{x}_k$ be the number of successive balls on the arc, then $|1-x_1|+|x_1-x_2|+\mathrm{......}+|x_k-9|\ge |1-9|=8$ equality occurs $\iff 1<x_1<x_2<\mathrm{....}<9$

no. of the balls on each route is increasing from 1 to 9 $\therefore S_{\min }=2\times 8=16$
Divide the set of 7 balls $\{2,3,4,\mathrm{....8}\}$ into two subsets, Then the subset which contains less elements has ${ }^7{C}_0{+}^7{C}_1{+}^7{C}_2{+}^7{C}_3=2^6=64$ . if clockwise and anticlockwise arrangement are different then answer will be 64*2=128