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

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

• Suppose every pair of lines intersects. How many times can two lines intersect? How many intersections could there be in all? – hardmath Jun 22 '14 at 14:40

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