Combinations Using the Circumference of A Circle

Well, first of all, ignoring the type of triangle formed, how many combinations total? The easiest way to think about this is to use the Fundamental Counting Principle. For the first dot, 15 choices, then 14 left for the second choice, then 13 left for the third choice: that’s 15 14 13. But, that will count repeats: the same three dots could be chosen in any of their 3! = 6 orders, so we have to divide that number by 6.

$\frac{15*14*13}{6}$

Cancel the factor of 3 in 15 and 6

$\frac{5*14*13}{2}$

Cancel the factor of 2 in the 14 and 2

$(5*7*13) = 5*91 = 455$

That’s how many total triangles we could create.

Of these, how many are equilateral triangles? Well, the only equilateral triangles would be three points equally spaced across the whole circle. Suppose the points are numbers from 1 to 15. From point 1 to point 6 is one-third of the circle — again, from point 6 to point 11, and from point 11 back to point 1. That’s one equilateral triangle. We could make an equilateral triangle using points {1, 6, 11}

{2, 7, 12}

{3, 8, 13}

(4, 9, 14)

{5, 10, 15} After that, we would start to repeat. There are five possible equilateral triangles, so 455 – 5 = 450 of these triangles are not equilateral. Therefore, the answer is $\boxed{450}$ .