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Intersecting Lines

Two lines in the same plane can have zero points of intersection (if parallel) or one point of intersection (if they cross). Three lines in the same plane can have zero (if all three lines are parallel), one (if all three lines intersect at the same point), two (if two lines are parallel and the third is a transversal that crosses both of them), or three (if the lines form the sides of a triangle) points of intersection. Suppose a total of nineteen different lines are arranged in the same plane. Determine how many different possibilities exist for the number of points of intersection of the nineteen lines.

(Source: Mathematics Teaching in the Middle School, Nov-Dec 1995)

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