AUSTRIAN-POLISH 1985 Problem 8

A convex $n$ -gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$ -gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$ , for $i=1,2,\dots,n-2$ . Find the number of all such labellings.