Division of m+n distinct object into two group of size m and n

Division of objects into group of equal size

Division Into Groups | Distinct Objects

Division into groups | Alike Objects

The number of ways of arranging n(gt2) distinct objects in a line so that two particulars objects are never together is

The number of ways of arranging n(gt2) distinct objects in a line so that two particulars objects are never together is

Solve the any one of following sub questions: If m and n are two distinct numbers then prove that m^(2) - n^(2), 2mn and m^(2) + n^(2) is a pythagorean triplet.