Final answer:
The number of triangles that can be formed using diagonals from a common vertex in a 17-gon is 91, calculated by selecting 2 vertices from the remaining 14 vertices to form a triangle with the starting vertex.
Step-by-step explanation:
To determine the number of triangles that can be formed using diagonals from a common vertex in a 17-gon (a polygon with 17 sides), consider that a triangle needs three points to be defined. When counting triangles in a polygon, you usually select two additional points along with your vertex to form a triangle. Since we have a 17-gon, we ignore the common vertex itself and the adjacent vertices (because connecting them doesn't form a diagonal but a side of the polygon), leaving us with 14 other vertices to work with.
To form a triangle, we need to select 2 vertices from these remaining 14, and the number of ways to do this is given by the binomial coefficient C(14, 2), which is the same as 14 choose 2. This is calculated as 14! / (2! * (14-2)!), which simplifies to 14 * 13 / 2, resulting in 91 triangles.