How many triangles can be formed by choosing any 3 vertices of a 12-sided polygon (no three chosen vertices are collinear except at the polygon vertices)?

Introduction / Context:Any three non-collinear points determine a triangle. In a convex (or general position) polygon, any 3 distinct vertices form a valid triangle.

Given Data / Assumptions:

• Polygon has 12 vertices.
• No three vertices are collinear beyond polygon vertices.
• All 3-vertex choices produce distinct triangles.

Concept / Approach:Choose any 3 of the 12 vertices using combinations.

Step-by-Step Solution:Number of triangles = C(12,3) = 12*11*10 / (3*2*1) = 220.

Verification / Alternative check:For an n-gon, the general formula is C(n,3). Substituting n=12 yields 220.

Why Other Options Are Wrong:200 and 240 are common arithmetic slips; 260 is too high.

Common Pitfalls:Subtracting “diagonals” or internal structures unnecessarily; triangles only depend on 3 vertices.

Final Answer:220