Question 8
A rectangular piece of paper measures $4$ units by $5$ units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called *basic* if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle.
Given that the total length of all lines drawn is exactly $2007$ units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by $1000.$
Given that the total length of all lines drawn is exactly $2007$ units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by $1000.$