Topics
Given n points in a plane, out of which k are collinear, how many lines can be formed by joining pairs of these points?
This is a classic combination problem. Each line is uniquely determined by choosing 2 points out of n , i.e. C(n, 2). From this, we subtract the overcounting due to the k collinear points: C(k, 2). Finally we add 1 for the line that contains all the k points. Thus, C(n,2)-C(k,2)+1.
If we have many such collinear “groups”, say G, within n. Then, tootal: C(n,2), out of which we subtract overcounting from these groups: C(G1, 2) + C(G2, 2) ... C(Gk, 2) and add 1 for the single lines contributed from each group: 1 + 1 ... 1 = k. Overall, ans:
Tip
Similar logic is applied for counting triangles as well. Only difference is that we don’t add 1 since collinear points don’t contribute a triangle (but they do contribute a line). Thus, we have: