BTGmoderatorLU wrote:Source: Veritas Prep
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
E. 84
To form a triangle, we must select 3 points such that at most 2 are collinear.
Case 1: Select 3 points not on the x-axis
From the 4 points not on the x-axis, the number of ways to choose 3 = 4C3 = (4*3*2)/(3*2*1) = 4.
Case 2: Select 1 point on the x-axis and two points not on the x-axis
From the 5 points on the x-axis, the number of ways to choose 1 = 5C1 = 5.
From the 4 points not on the x-axis, the number of ways to choose 2 = 4C2 = (4*3)/(2*1) = 6.
To combine these options, we multiply:
5*6 = 30.
Case 3: Select 2 points on the x-axis and 1 point not on the x-axis
From the 5 points on the x-axis, the number of ways to choose 2 = 5C2 = (5*4)/(2*1) = 10.
From the 4 points not on the x-axis, the number of ways to choose 1 = 4C1 = 4.
To combine these options, we multiply:
10*4 = 40.
Total ways = Case 1 + Case 2 + Case 3 = 4 + 30 + 40 = 74.
The correct answer is
C.
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