[EM] number of possible ranked ballots given N candidates

[Paul Kislanko]

James Gilmour: wrote
> 
> Maybe there are 4 million possibilities with 10 candidates, 
> but you won't have 4 million actual combinations unless you
> have many more voters than 4 million.

I was going to mention this as well. From a practical standpoint, one only
need record the "forms" of ballots actually cast, then for each new ballot
see if it's "form" is already stored, and if so just add 1 to the number of
voters who used that "form". 

> 
> In the STV-PR 2002 election in Meath (Ireland) for which the 
> full ballot "papers" are available electronically, there
> were 14 candidates for 5 places, so there were about 2.4 
> ×10^11 possible combinations of preferences.  However, there
> were only 64,081 voters, so the practical maximum was 64,081. 
>  But in fact, those 64,081 voters marked only 25,101
> unique sets of preferences.  The most common preference list 
> appeared on 1,618 ballots.
> 
> Of course, this won't help if you are trying to develop an 
> "all possible combinations matrix".

Just ininitializing an "all possible configurations" could be daunting. But
the method I described above can be done efficiently enough to run on a
laptop in a few seconds.