The number of ways the 2 chosen students are from Grade 10, North $={}_{47}{C}_2=\frac{47!}{2!}$

The number of ways the 2 chosen students are from Grade 10, South = ${}_{93}{C}_2=\frac{93!}{2!}$

The number of ways the 2 chosen students are from Grade 11, North = ${}_{73}{C}_2=\frac{73!}{2!}$

The number of ways the 2 chosen students are from Grade 11, South = ${}_{57}{C}_2=\frac{57!}{2!}$

The total number of ways 2 students are chosen = ${}_{270}{C}_2=\frac{270!}{2!}$