The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Solution

Maths sir · 4 steps

Count all triangles, then remove the ones that use one or two sides of the octagon.

1. All triangles. No three vertices of a regular octagon are collinear, so any 3 of the 8 vertices give a triangle: C(8, 3) = 56.
2. Two octagon sides. A triangle uses two sides of the octagon when its vertices are three consecutive vertices. There is one such triple starting at each vertex, so there are 8 of these.
3. Exactly one side. Pick one side of the octagon in 8 ways. The third vertex cannot be either endpoint or one of the two vertices next to them, so it has 8 - 4 = 4 choices. That gives 8 × 4 = 32 triangles.

   56 - 8 - 32 = 16

4. So. There are 16 triangles with no side along the octagon, which is option (b).

The idea to keep. To count triangles with no side of an n-gon, subtract those with two sides (n) and exactly one side (n(n-4)) from C(n, 3). That leaves n(n-4)(n-5)/6.

Answer from the NTA final answer key. Solution by PrepWiser.