Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:

Count the number of triangles with positive area whose vertices are points whose $(x, y)$ -coordinates lie in the set $\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)\}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

There are $\binom{9}{3} = 84$ triples of points. 8 of them form degenerate triangles (the ones that lie on a line), so there are $84 - 8 = 76$ nondegenerate triangles.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.