Mind Blown!?!?!?Number Theory Level 5
Among \(100\) points in a plane, no three collinear, exactly \(4026\) pairs are connected by line segments. Each point is then randomly assigned an integer from \(1\) to \(100\) inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least \(50\).