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

Step-by-step solution

The number of triangles formed from vertices of a regular 20-gon without using polygon sides equals the count of triangles that share no sides with the polygon. The total number of triangles is given by C(20,3) = 20 19 18 3 2 1 = 1140 . Triangles sharing exactly two sides correspond to three consecutive vertices, yielding 20 such triangles. Triangles sharing exactly one side are formed by selecting one side ( 20 choices) and a non-adjacent third vertex ( 20 - 4 = 16 choices), giving 20 16 = 320 triangles. The count