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Why is the maximum number of intersections of k lines in $\mathbb{R}^2$ = $\binom{k}{2}$?

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  • $\begingroup$ Suppose every pair of lines intersects. How many times can two lines intersect? How many intersections could there be in all? $\endgroup$ – hardmath Jun 22 '14 at 14:40
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Consider two lines : the max number of intersections is 1.
Then the third line will intersect at max the two others lines so the max number of intersection is 1+2=3.
Then the fourth line will intersect at max the three others lines so the max number of intersection is 1+2+3=6.

By induction, it is therefore easy to show that the maximum number of intersections is $$\sum_{i=1}^{k-1} i = \frac{(k-1)k}{2} = \binom{k}{2}$$

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Each pair of line can intersect each other once (as far as they aren't the same line) and there is $\binom{k}{2}$ pairs among k lines.

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