A rock-paper-scissors tournament is being held for players from different regions

Question

A rock-paper-scissors tournament is being held for players from different regions -
North, East and South. In the first round, each player competes against all other
players not from their own region exactly once.
Part way through the first round, players from the West arrive. In order to not disrupt
the games being played amongst the other regions, players from the West compete with
each other exactly once.
How many games of rock-paper-scissors are played in this first round if there are 5
players from the North, 7 from the East, 8 from the South and 15 from the West?

Answer 1

Hi, Ryan:

Here's how to do this one:

Initially, we have players from N (5 players), E (7) and S (8).

Each N player plays 7 + 8 = 15 games for a total of 5 x 15 = 75 games

Each E player plays against N, which we've already counted, and against S for a total of 7 x 8 = 56 games

Each of the 15 W players plays against his/her other 14 team mates.

So, the 1st W player, plays 14 games.

The 2nd W player plays 13 more games,

And so on, for a total of 15 + 14 + 13 + . . . + 2 + 1 = 15 . 16/2 = 120 games

Total of all games played is 75 + 56 + 120 = 251 games.