Combinations n sided polygon Three vertices of a convex n sided polygon are selected. If the number of triangles that can be constructed such that none of the sides of the triangle is also the side of the polygon is 30 then 'n' is

Assuming $n\ge 5$:
Total number of triangles:
$$\binom n3$$
Triangles such that exactly one of its sides is a side of the polygon:
$$n(n-4)$$
Triangles such that exactly two of its sides are sides of the polygon:
$$n$$
Then
$$\binom n3-n(n-4)-n=30$$
$$n^3-9n^2+20n-180=0$$
$$(n-9)(n^2+20)=0$$
