Bunuel wrote:

Set S contains nine distinct points in the coordinate plane. If exactly five of the points lie on the x axis, and if no other set of three points in S is collinear, how many triangles can be formed by taking points in S as vertices?

A. 56

B. 70

C. 74

D. 79

D. 84

We have 5 (collinear) points that are on the x-axis and 4 points that are not on the x-axis. Since no set of three points in S is collinear except those on the x-axis, we can have the following 3 cases forming a triangle: 1) two points on the x-axis and one point not on the x-axis, 2) one point on the x-axis and two points not on the x-axis, and 3) three points not on the x-axis.

Case 1: Two points on the x-axis and one point not on the x-axis

There are 5C2 x 4C1 = 10 x 4 = 40 such triangles in this case.

Case 2: one point on the x-axis and two points not on the x-axis

There are 5C1 x 4C2 = 5 x 6 = 30 such triangles in this case.

Case 3: Three points not on the x-axis

There are 4C3 = 4 such triangles in this case.

Therefore, there are 40 + 30 + 4 = 74 triangles that can be formed.

Answer: C

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