Question 27
Let $n$ be a given integer with $n$ greater than $7$, and let $\mathcal{P}$ be a convex polygon with $n$ sides. Any set of $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n-2$ triangles. A triangle in the triangulation of $\mathcal{P}$ is an interior triangle if all of its sides are diagonals of $\mathcal{P}$.
Express, in terms of $n$, the number of triangulations of $\mathcal{P}$ with exactly two interior triangles, in closed form.
Express, in terms of $n$, the number of triangulations of $\mathcal{P}$ with exactly two interior triangles, in closed form.