Can we uniquely using matrix answers?

I'm setting up a form for a judges' panel for a contest we're holding. We have 16 entries, and we need to narrow it down to 5.

I'd like my judges' to select their top 5 entries (and want them to specify 1st place, 2nd place, etc.), and they can use each place only once. I was hoping to use a matrix table (with 1st, 2nd, etc. in the columns, and the names of the entrants in the rows) so that the judges could see all 16 entrants at one time, rather than doing individual drop-downs for the places. But I don't see how to make the table restrict the number of times you can select a column. 

Is there a way to program the form to allow only one entry per column?

We cannot limit the answers within Matrix field, but we have a widget with sortable list of items which seems to be suitable in this case.

Please take a look at the next widget.

Orderable list 

Thank you.