Question 15
Given a positive integer $n$, $2n$ aligned points in a plane are considered $A_1, A_2, \dots, A_{2n}$. Each point is colored blue or red using the following procedure: On the given plane, $n$ circles with end diameters $A_i$ and $A_j$ are drawn, disjointed two by two. Each $A_k$, $1 \le k \le 2n$, belongs to exactly one circle. The points are colored so that the two points on the same circle have the same color.
Determine how many different colorations of the $2n$ points can be obtained by varying the $n$ circumferences and distribution of the two colors.
Determine how many different colorations of the $2n$ points can be obtained by varying the $n$ circumferences and distribution of the two colors.