Darts Tournament with Eight Players

Date: 10/05/2002 at 22:53:46
From: Chad Lakin
Subject: Story problem

There are eight players in a darts tournament. Each player plays one 
game against each of the other players. How many dart games will be 
played in the tournament? Find a pattern to determine the number of 
games played no matter how many players there are.  

The number is 28 but I don't know how to get the mathamaticical 
formula.

Date: 10/06/2002 at 11:05:50
From: Doctor Ian
Subject: Re: Story problem

Hi Chad,

Let's look at a simpler tournament, in which there are only three 
players: Pat, Chris, and Jody.

Suppose we're going to set up a match at random, by choosing a pair of 
competitors. How many ways can we choose the first competitor? There 
are three possible choices, right? 

  1. Pat and _____
  2. Chris and ____
  3. Jody and ____

Now, in how many ways can we choose the second competitor? We have 
only 2 choices.  

  Pat and _____    ->  1.1. Pat and Chris
                       1.2. Pat and Jody
  Chris and ____   ->  2.1. Chris and Pat
                       2.2. Chris and Jody
  Jody and ____    ->  3.1. Jody and Pat
                       3.2. Jody and Chris

So the number of pairings we could have is 3*2, or 6:

But wait! Choosing Pat and Chris is the same thing as choosing Chris 
and Pat, right? So there are _really_ only half as many choices as we 
thought: (3*2)/2, or 3.  

Can you use the same logic to come up with 28 pairings for 8 
players? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/