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by ygdrasil24 » Tue Aug 06, 2013 7:31 pm
10. 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
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by GMATGuruNY » Tue Aug 06, 2013 7:40 pm
ygdrasil24 wrote:10. 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
A triangle is a combination of 3 points. Since 1≤x≤3 and 1≤y≤3, we have 9 points from which to choose:

Image

The number of combinations of 3 that can formed from 9 choices = 9C3 = 84.

But some of these combinations will result in a straight line. Using the 9 points, 8 lines can be formed:

Image

Thus, we need to subtract these 8 lines from our total:

Total possible triangles = 84-8 = 76.

The correct answer is B.
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