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
Real simply- the wording in this question in saying that 5 points have an x value of zero and therefore line on the x-axis therefore these five points are collinear and cannot form a triangle. Secondly, when this question states no other set of three points in S is collinear- all that really means is that of the 4 points left these four points cannot form a straight line.
Number of Triangles Possible without restriction- also in order to understand the subtraction of the restriction consider this example. Freddie, Shaggy, Velma, Daphne, and Scooby are about to hunt down a ghost but only two members of can be chosen for the hunt. But Freddie and Daphne just got into an argument so they refuse to go together. So the total number of combinations that could be made, and remember order doesn't matter, would be 5c2-2c2=9. When we say order doesn't matter well what that means is Freddie and Scooby in a group represent one possibility- it doesn't matter what position they are in FS or SF because that constitutes the same combination. If we just did 9c3 we would count the total number of possible triangle that include triangles made from the five points.
9c3 - 5c3 = 74
Thus
"C"
*Notice- on the diagram we cannot form a straight line with 3 points that include any of the points (I, II, III, IV)
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