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

The Correct Option is C

To solve the problem of finding 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, we can follow these steps:

First, calculate the total number of triangles that can be formed by choosing any three vertices from the octagon. A regular octagon has 8 vertices. The number of ways to choose 3 vertices from 8 is given by the combination formula:

\(^8C_3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\).

Next, subtract the number of triangles where at least one side is a side of the octagon from the total number of triangles.

To determine these triangles, notice that for a triangle to have a side of the octagon, it must include two adjacent vertices:

• Once two adjacent vertices are chosen, the third vertex can be chosen from the remaining vertices that do not share an octagon side with either of the two chosen vertices.
• In a regular octagon, choosing two adjacent vertices results in choosing the third vertex from the remaining 5 vertices not adjacent to the first two.

There are 8 such pairs of adjacent vertices (one pair for each side of the octagon). Thus, the number of unwanted triangles (where one side is a side of the octagon) is:

\(8 \times 5 = 40\).

Finally, subtract the number of unwanted triangles from the total number possible:

\(56 - 40 = 16\).

Thus, the number of triangles where none of the sides is a side of the octagon is 16. 
Therefore, the correct answer is 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.