Question

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

• A. 24
• B. 56
• C. 16
• D. 48

Answer 1

Answer: (C)

16

Answer 2

The number of triangles having no side common with an $n$-sided polygon is given by:

$\text{no. of triangles having no side common with a } n \text{-sided polygon} = \binom{n}{1} \times \binom{n-4}{2} \div 3$

Substitute $n = 8$:

$= \binom{8}{1} \times \binom{4}{2} \div 3$

$= 8 \times 6 \div 3$

$= 16.$

Thus, the number of such triangles is 16.