SAT Math problem

Question

In a volleyball league with 4 teams, each team plays exactly 2 games with each other 3 teams in the league. What is the total number of games played in this league?

the book says the answer is 12, i wondered how come is that?

Answer 1

Team A can play 3 games with each of the three teams.

Team B can play 3 games too , but it's game with Team A has already been counted so we leave that out. So we count 2 games.

Team C can play 3 games too , but it's game with Team A and B has already been counted so we leave those out.So we count 1 game.

Team D can play 3 games too , but it's game with Team A and B and C has already been counted so we leave those out.So we count none.

So total games =3*2*1=6

If they play two games with each other then total games is 6*2=12 .

You can count these things faster after learning Permutation and Combination.

Answer 2

This is similar to handshake problem :

Choosing $2$ teams from the available $4$ teams gives you $\binom{4}{2} = 6$ games

Take twice of above since each team plays two games with every other team

Answer 3

Imagine that of the two games between any two teams X and Y, one is at the "home" of X, and the other is at the home of Y.

Each of the $4$ teams plays $3$ home games. Since any game is a home game for one of the teams, there are $(4)(3)$ games.

Answer 4

If your having trouble visualizing the answer given above by ganeshie8. Just draw 2 connections between each pair of teams, you'll have 12 connections in the end.