Question

A regular polygon has 20 sides. The number of triangles that can be drawn by using the vertices but not using the sides is

Options

• 1140

• 800

• 340

• 20

Answer 1

800

Explanation:

Total triangles = \[^{20}\mathbb{C}_3\]

Triangles which have two sides common with polygon = 20

Triangle which have one side common with the polygon = \[20\times{}^{16}\mathrm{C}_{1}\]

∴ Required number of triangles

\[={}^{20}\mathrm{C}_3-20-20\times{}^{16}\mathrm{C}_1\]

\[=\frac{20\times19\times18}{1\times2\times3}-20-16\times20\]

= 20 (57 − 1 − 16)

= 20 × 40

= 800