IR-TST 2012 Problem 17

Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$ . Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$ -gon (are in convex position), but the points of $A$ are not, prove that $T(B)<T(A)$ .

Proposed by Ali Khezeli