2023 AIME I Problems/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.

Solution

The number of points where exactly 2 lines intersect is $\begin{align*} & \binom{40}{2} - 3 \cdot \binom{3}{2} - 4 \cdot \binom{4}{2} - 5 \cdot \binom{5}{2} - 6 \cdot \binom{6}{2} \\ & = \boxed{\textbf{(607) }} . \end{align*}$

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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2023 AIME I Problems/Problem 3

Solution