Logic, Pattern, and Formula for maximum number of regions that can be made by intersection of n lines in a plane.
Questions:
What is the maximum number of regions created by intersection of 5 lines in a plane?
How many regions (maximum) will 6 intersecting lines create in a plane ?
One line creates 2 regions in a plane (whichever way we draw it)
Total regions =2
2nd line creates 2 more regions in the plane when it intersects with the 1st line.
Total regions = 2+2=4
For creating maximum number of regions , point of intersection of each pair of lines should be different.
3rd line can create maximum 3 more regions in the plane when it intersects with the
existing 2 lines.
Total regions = 3+2+2=7
4th line can create maximum 4 more regions in the plane when it intersects with the
existing 3 lines.
Total regions = 4+3+2+2=11
Similarly,
5th line will create total = 5+4+3+2+2 = 16 regions
6th line will create total = 6+5+4+3+2+2 =22 regions
Hence, every nth line can create maximum n number of extra regions in a plane when it intersects with the existing (n-1) lines.
Hence n lines will create maximum n+(n-1)+(n-2)……..3+2+2 regions
= n(n+1) + 1 Regions
2
