UKMT Senior Kangaroo, 2022

A class consists of pupils who each wear either a red tie or a green tie. There are two more pupils wearing green ties than wearing red ties. Two pupils are chosen at random. The probability of selecting two pupils of different tie-colour is exactly treble the probability of selecting two pupils who both wear red ties. Given that $R$ is the number of pupils wearing a red tie, and that $R > 0$, determine $R^3 + R^2 + R$.