Question

A team has 13 players, 5 of whom are on the field at any one time. How many different 5-player groups can be made?

Answer 1

1287

Explanation:

Combinations use this general formula:

`C_(n,k)=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

`C_(13,5)=(13!)/((5!)(13-5)!)=(13!)/((5!)(8!))`

To solve, you can use a table to find the different values (or a calculator), but factorials get quite big quite fast, and so I find it's often easier to simply work out the cancellations yourself:

`(13!)/((5!)(8!))=(13xxcancel12xx11xxcancel10xx9xxcancel(8!))/(cancel(5xx4xx3xx2)xxcancel(8!))=13xx11xx9=1287`

This shows the number of 5-player groups we can make.

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If we care about the order of which player goes first or about what position each player will take, then we'd want to use the permutation formula:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`