Final answer:
From a common vertex in a 20-gon, 136 triangles can be formed using diagonals, by calculating the combinations of 17 vertices (excluding the adjacent ones) taken 2 at a time.
Step-by-step explanation:
To calculate the number of triangles that can be formed using diagonals from a common vertex in a 20-gon, we follow a simple combinatorial rule. In a polygon with n sides, a triangle is determined by selecting any two nonadjacent vertices in addition to the vertex in question. Since we are forming triangles using a common vertex and diagonals, we cannot choose the two vertices directly adjacent to the common vertex.
Thus, in a 20-gon, there are 20 - 1 - 2 = 17 vertices that can be chosen to form a triangle with the common vertex. Selecting any two of these will give us a triangle, so we calculate the combinations of 17 vertices taken 2 at a time (17 choose 2).
To find the number of triangles that can be formed using diagonals from a common vertex in a 20-gon, we need to determine the number of ways we can choose 2 vertices out of 20. In a 20-gon, each vertex is connected to 18 other vertices via diagonals. So, we can choose 2 vertices from 20 in 20C2 ways, which is equal to 190.
Calculating the combination, we get:
17 choose 2 = 17! / (2! * (17 - 2)!) = 17 * 16 / (2 * 1) = 136.
Therefore, using diagonals from a common vertex, 136 triangles can be formed from a 20-gon.