Suppose in the plane 10 pair wise nonparallel lines intersect one...

Number of non-overlapping polygons $=56-20=46$ 1 line divide plane into 2 regions 2 lines divide plane into 4 regions 3 lines divide plane into 7 regions 4 lines divide plane into 11 regions 5 lines divide plane into 16 regions 6 lines divide plane into 22 regions 7 lines divide plane into 29 regions 8 lines divide plane into 37 regions 9 lines divide plane into 46 regions 10 lines divide plane into 56 regions Now open regions for 3 lines are 6
Similarly for 10 lines are 20 Note : If we consider overlapping polygons then maximum possible number of polygons $={ }^{10} \mathrm{C}_3+{ }^{10} \mathrm{C}_4+\ldots . .+{ }^{10} \mathrm{C}_{10}=2^{10}-1-10-45=968$