15 points

Geometry Level 4

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?


The answer is 395.

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1 solution

Áron Bán-Szabó
Jul 12, 2017

If we want to choose three points, we can do it ( 15 3 ) = 15 14 13 1 2 3 = 455 \binom{15}{3}=\dfrac{15*14*13}{1*2*3}=455 ways. The three points always define a triangle, except they are on the same line. There are three lines, and there are 6 6 points in each of them. Since three points can't be on two lines at the same moment, from the 455 455 ways 3 ( 6 3 ) = 3 6 5 4 1 2 3 = 60 3*\binom{6}{3}=3*\dfrac{6*5*4}{1*2*3}=60 ways aren't make solutions.

Therefore the answer is 455 60 = 395 455-60=\boxed{395} .

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