Problem:
Ten points are selected on the positive -axis, , and five points are selected on the positive -axis, . The fifty segments connecting the ten selected points on to the five selected points on are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?
Answer Choices:
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Solution:
A point of intersection in the first quadrant is obtained whenever two of the segments cross to form an . An is uniquely determined by selecting two of the points on and two of the points on . The maximum number of these intersections is obtained by selecting the points on and so that no three of the segments intersect in the same point. Therefore, the maximum number of intersections is .
Choose ten points, , on . Choose on and draw the ten segments joining to the ten points on . Choose on , and note that, as the segments are drawn, intersections are formed. Choose on so no segment goes through a previously counted intersection, and note that new intersections are formed. Similarly, for judiciously chosen and on one can generate at most and new intersections, respectively. Hence, the maximum number of intersections is intersections.