What hammock to choose?

The maximum number of regions formed by n lines that intersect inside a circle is (n2)+(n1)+(n0)=(n+12)+1(n2)+(n1)+(n0)=(n+12)+1{{n}\choose{2}}+{{n}\choose{1}}+{{n}\choose{0}}={{n+1}\choose{2}} +1. What is an intuitive explanation for these formulas that a 12-year-old can understand?

• I.e. what n choose 2 "objects" are we choosing, what n choose 1 objects are we choosing, what n+1 choose 2 objects are we choosing?

• Answer:

  Say we have divided a circle into some number of regions with nnn lines. In each region, choose the lowest point. (If this description is ambiguous, rotate the diagram until it is not.) Each region’s lowest point is either a vertex between two lines, a vertex between the circle and one line, or the lowest point of the circle on zero lines. Furthermore, any set of two, one, or zero lines uniquely identifies at most one region in this way. Therefore the number of regions is bounded by (n2)+(n1)+(n0)(n2)+(n1)+(n0)\tbinom n2 + \tbinom n1 + \tbinom n0. In the case that all the intersection points are distinct, each set of two, one, or zero lines identifies exactly one region, so this maximum is achieved. This generalizes to any number of dimensions: for example, nnn planes divide a sphere into at most (n3)+(n2)+(n1)+(n0)(n3)+(n2)+(n1)+(n0)\tbinom n3 + \tbinom n2 + \tbinom n1 + \tbinom n0 volumes.