Question-58

A round-robin tournament is one in which every player plays against every other player exactly once. Consider a round-robin tournament with \(\displaystyle n\) players in which each game can result in either a win or a loss (no draws are possible). Define an outcome-list of a tournament to be the list of the names of winners and losers for each game.

• The total number of games played is \(\displaystyle 2^{n}\)

• The total number of games played is \(\displaystyle \binom{n}{2}\)

• The total number of outcome-lists for a specific tournament is \(\displaystyle 2^{n}\)

• The total number of outcome-lists for a specific tournament is \(\displaystyle 2^{\binom{n}{2}}\)

Answer

• The total number of games played is \(\displaystyle 2^{n}\)

• The total number of games played is \(\displaystyle \binom{n}{2}\)

• The total number of outcome-lists for a specific tournament is \(\displaystyle 2^{n}\)

• The total number of outcome-lists for a specific tournament is \(\displaystyle 2^{\binom{n}{2}}\)

Solution

There are \(\displaystyle n\) choices for the first player and \(\displaystyle n-1\) choices for the second player. Using the multiplication rule, we have \(\displaystyle n( n-1)\) choices for a pair of players. However, a game is an unordered pair, so we have \(\displaystyle n( n-1) /2\) games in total. Alternatively, a game is a choice of two players from the set of \(\displaystyle n\) players. The total number of games is given by \(\displaystyle \binom{n}{2}\).

Given a tournament, each game has two possibilities. Therefore, the total number of outcome-lists would be \(\displaystyle 2^{\binom{n}{2}}\).