If you draw all possible diagonals from a single vertex of a regular $decagon$, how many distinct triangles can be formed using that vertex and two other vertices of the $decagon$?

• A. $8$
• B. $20$
• C. $36$
• D. $28$

Answer: (C)

Verified step by step guidance

1. Identify the number of vertices in a decagon, which is 10.
2. Since we are focusing on one specific vertex, note that this vertex will be included in every triangle formed.
3. To form a triangle using this chosen vertex and two other vertices, we need to select 2 vertices from the remaining 9 vertices of the decagon.
4. Use the combination formula to find the number of ways to choose 2 vertices from 9: \(\displaystyle \binom{9}{2} = \frac{9!}{2! \times (9-2)!}\).
5. Calculate the value of \(\binom{9}{2}\) to find the total number of distinct triangles that can be formed using the chosen vertex and two other vertices.