How many triangles?

Probability Level 3

How many non-degenerate triangles can you obtain choosing three points in the picture below as vertices?


The answer is 76.

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Matteo Benigni by Matteo Benigni , 21, Italy

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1 solution

Matteo Benigni
Nov 24, 2016

There are ( 9 3 ) {9 \choose 3} possible triangles, and we need to subtract the number of degenerate triangles (there are 8 8 of them, since we have 3 rows, 3 colums and 2 diagonals, all of which are made form 3 points that are lined up, so the result is a degenerate triangle): 9 ∗ 8 ∗ 7 3 ∗ 2 − 8 = 76 \frac{9*8*7}{3*2} - 8 = \boxed{76}

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How do you know that there are 8 of them?

Pi Han Goh - 4 years, 6 months ago

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I should include that in the answer, thank you

Matteo Benigni - 4 years, 6 months ago

Um, The degenerate triangles yes I think there are just 8, 3 vertical, 3 horizontal and two diagonal.

Ed Sirett - 4 years, 6 months ago

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