AUSTRIAN-POLISH 2005 Problem 1

For a convex $n$ -gon $P_n$ , we say that a convex quadrangle $Q$ is a diagonal-quadrangle of $P_n$ , if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$ . Let $d_n$ be the number of diagonal-quadrangles of a convex $n$ -gon. Determine $d_n$ for all $n\geq 8$ .