On a circle there are 15 distinct points; how many different triangles can be formed by joining these points as vertices?

Introduction / Context:
This problem checks your understanding of combinations and basic counting principles. When you form triangles from points on a circle, you simply need to count how many ways you can select 3 distinct points out of the total points, because any three non-collinear points form exactly one unique triangle.

Given Data / Assumptions:

• There are 15 distinct points on a circle.
• No three points are collinear except on the circle itself, so any 3 points chosen form a valid triangle.
• We are counting triangles only by their vertex sets, not by drawing order or orientation.

Concept / Approach:
To form a triangle, you must choose 3 vertices from the 15 available points. The order in which you choose these 3 vertices does not matter; hence we use combinations, not permutations. The relevant formula is nC3 = n * (n - 1) * (n - 2) / 6 for the number of ways to choose 3 items from n items without regard to order.

Step-by-Step Solution:
We need to compute 15C3.Use the formula: 15C3 = 15 * 14 * 13 / 6.First compute the numerator: 15 * 14 = 210, and 210 * 13 = 2730.Now divide by 6: 2730 / 6 = 455.Therefore, the number of distinct triangles = 455.

Verification / Alternative check:
We can simplify during multiplication: 15C3 = (15 * 14 * 13) / (3 * 2 * 1).Cancel 14 with 2 to get 7, and cancel 15 with 3 to get 5.Now compute 5 * 7 * 13 = 35 * 13 = 455.This quick simplification confirms the calculation.

Why Other Options Are Wrong:
450 is close to the correct value and could come from rounding or subtracting 5 by mistake.469 and 500 do not correspond to any simple combination calculation with 15 points and are clear overestimates.

Common Pitfalls:
Using permutations (15P3) instead of combinations, which would incorrectly count each triangle multiple times.Wrongly subtracting combinations or assuming some triangles are degenerate, which is not the case for distinct points on a circle.Careless arithmetic when multiplying or dividing large numbers.

Final Answer:
The number of triangles that can be formed is 455.