Math Gold Medalist

Lor

2023 AIME I 

Problem 3

A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Double Counting

40C2 = k . 2C2 + 3 . 3C2 + 4 . 4C2 + 5 . 5C2 + 6 . 6C2 

   Solution