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Lines and intersection

Source: Nigeria Olympiad senior mathematics round 2 2020 problem 3

January 19, 2020
combinatorics

Problem Statement

NN straight lines are drawn on a plane. The NN lines can be partitioned into set of lines such that if a line ll belongs to a partition set then all lines parallel to ll make up the rest of that set. For each n>=1n>=1,let ana_n denote the number of partition sets of size nn. Now that NN lines intersect at certain points on the plane. For each n>=2n>=2 let bnb_n denote the number of points that are intersection of exactly nn lines. Show that ∑n>=2(an+bn)\sum_{n>= 2}(a_n+b_n)(n2)\binom{n}{2} == (N2)\binom{N}{2}
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