Each of an equilateral triangle's sides is divided into five equal parts.
How many triangles are there, such that all of their vertices are from the 15 points?
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If we want to choose three points, we can do it ( 3 1 5 ) = 1 ∗ 2 ∗ 3 1 5 ∗ 1 4 ∗ 1 3 = 4 5 5 ways. The three points always define a triangle, except they are on the same line. There are three lines, and there are 6 points in each of them. Since three points can't be on two lines at the same moment, from the 4 5 5 ways 3 ∗ ( 3 6 ) = 3 ∗ 1 ∗ 2 ∗ 3 6 ∗ 5 ∗ 4 = 6 0 ways aren't make solutions.
Therefore the answer is 4 5 5 − 6 0 = 3 9 5 .