There are 4C2 = 6 ways to select 2 schools.
Given those 2 schools there are 42 = 16 ways to pair up players 1 from each school
There are 3 games played by each of those 3 pairs.
This is a total of 6 x 16 x 3 = 288 games played against other schools.
There are 4C2 = 6 ways to select a pair from the 4 people at a given school.
There are 4 schools
They play 1 game each
This is a total of 6 x 4 x 1 = 24 games played against schoolmates
A total of 288 + 24 = 312 games