S is a set of points in the plane. How many distinct

[BTGmoderatorDC]

S is a set of points in the plane. How many distinct triangles can be drawn that have three of the points in S as vertices?

(1) The number of distinct points in S is 5.
(2) No three of the points in S are collinear.

OA C

Source: Official Guide

[Ian Stewart]

We clearly need to know how many points we have, since if we have, say, only 2 points, we can't draw any triangles, but if we have many points that aren't all in a line, we can draw at least one triangle. So Statement 1 is indispensable. But knowing we have exactly 5 points is not sufficient, since if all of the points lie on a single line, connecting three of them will make a line, not a triangle. Using both statements, we know we have 5 points, and picking any 3 of them will always let us draw a triangle, since no 3 points ever form a straight line. So the number of triangles we can make is equal to the number of sets of 3 we can pick from a set of 5. Since this is DS, you don't need to even know how to calculate that, but that number equals (5)(4)(3)/3! = 10, and the answer is C.