An election ballot asks voters to select five city commissioners from a group of thirteen candidates. In how many ways can this be done?

This can be solved using permutations. You can get more into this online or just accept a quick answer here.

Since there are certain roles to be filled this is a permutation.

You need a somewhat smart calculator for this.

If I remember this right you'd type in the number of people you are choosing from, P, and then the places you are trying to fill.

So it should look like this:

13 P 5

You can find where permutations and combinations are in your calculator using a handbook or the internet but usually it looks like nPr for permutation or nCr for combination.