EXAMPLE 41 How many combinations in the intramural tennis league?

We return to the intramural singles tennis league at the local college. There are five players: Ryan, Megan, Nicole, Justin, and Kyle. Each player must play each other once.

  1. Confirm that a match between two players represents a combination.
  2. How many matches will be held?

Solution

  1. Let {Ryan, Megan} denote a tennis match between Ryan and Megan. Note:

    • There are players chosen from players.
    • Each player plays each other player once, so repetition is not allowed.
    • There is no difference between {Ryan, Megan} and {Megan, Ryan}, so order is not important.

    Thus, a tennis match between two players represents a combination.

  2. The list of all matches is as follows.
{Ryan, Megan} {Megan, Nicole} {Nicole, Justin}
{Ryan, Nicole} {Megan, Justin} {Nicole, Kyle}
{Ryan, Justin} {Megan, Kyle} {Justin, Kyle}
{Ryan, Kyle}

Thus, there are possible matches of players chosen from players.