There are n sides of a polygon(where $n>5$).
Triangles are formed by joining the vertices of the polygon. How many triangles can be constructed with no side common to the polygon?
My try:
Total possible triangles = $\frac{(n)(n-1)(n-2)}{6} = \binom{n}{3}$ ------(1)
Triangles with 2 sides common = $n$ ------(2)
Triangles with 1 side common = $n(n-4)$ ------(3)
So, with no side common = $1-2-3$
Is there any other way to get it directly without following this process?