Probability - Gravner-15.a

Question

A tennis tournament has $2n$ participants, $n$ Swedes and n Norwegians. First, $n$ people are chosen at random from the $2n$ (with no regard to nationality) and then paired randomly with the other $n$ people. Each pair proceeds to play one match. An outcome is a set of $n$ (ordered) pairs, giving the winner and the loser in each of the $n$ matches.

(a) Determine the number of outcomes.

Answer 1

The total orders will be $2n \choose n \times n!$

$2n \choose n$ is the number of ways of choosing the first player. After that, we have $n$ spots vacant - one corresponding to each player and these can be filled in $n!$ ways.