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triangles on coordinate plane - please elaborate

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triangles on coordinate plane - please elaborate

by gmatpup » Wed Nov 09, 2011 1:46 pm
How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1≤x≤3 and 1≤y≤3?

A. 72

B. 76

C. 78

D. 80

E. 84

Can someone please explain this step-by-step and also the logic used behind it?

Thanks so much!! :)
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by neelgandham » Wed Nov 09, 2011 2:18 pm
We have exactly 9 points (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3) to use as vertices.
We can select 3 points from these 9 points in 9C3 ways = 9!/(6!*3!) = 84, of which 8 sets of points result in straight lines. So, 84-8 = 76 triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1≤x≤3 and 1≤y≤3?

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