Some of the regions defined by n lines in the plane are infinite, while others are bounded. What’s the maximum possible number of bounded regions?

Question

topic: Lines in the plane from concrete mathematics

Answer 1

1 line splits the plane into 2 unbounded regions, but no bounded ones.

2 lines that are not parallel produce no bounded regions.

3 lines are needed to form a single bounded triangle. 1=(3-2)(3-2+1)/2.

4 lines can form 3 bounded regions when the 4th line intersects the triangle formed by the other three lines and forms a new region by intersecting one of the other lines. 3=1+2=(4-2)(4-2+1)/2.

5 lines can form 6 bounded regions. 6=1+2+3=(5-2)(5-2+1)/2.

6 lines can form 10 bounded regions. 10=1+2+3+4=(6-2)(6-2+1)/2.

n lines can form (n-2)(n-1)/2 bounded regions, where n>2.