How many triangles are formed by the vertices of a polygon having 15 sides?

Concept: For a convex polygon in which no three vertices are collinear, every set of 3 distinct vertices determines exactly one triangle. So the number of triangles formed by the vertices of an n-sided polygon equals the number of ways to choose 3 vertices from n, given by the combination formula C(n,3) = n(n-1)(n-2) / 6.

Application: Here n = 15, so the number of triangles is C(15,3).

1. Set up the combination: C(15,3) = 15 x 14 x 13 / (3 x 2 x 1)

2. Multiply the numerator: 15 x 14 x 13 = 2730

3. Divide by the denominator: 2730 / 6 = 455

Cross-check: Simplify before multiplying fully -- 15 / 3 = 5, so C(15,3) = 5 x 14 x 13 / 2 = 5 x 91 = 455, the same result via an independent route.

Conclusion: The polygon contains 455 distinct triangles formed by its vertices.