Nine balls numbered $1,2,3,\mathrm{......9}$ 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 neighbouring balls. The number of arrangements which make ‘s’ the minimum is

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