Do you need total precision? Using one of the 'BigMath' modules would allow you to get complete accuracy, but calculating large factorials will consume large amounts of memory and take a long time. Using one of the C-based math packages will calculate a lot faster but will discard some accuracy. By way of example, the following both produce the same results to 30+ significant digits, which as your probabilities will be in the range 0->1 or percentanges, you're probably only interested in 2 or 3 digits of accuracy in the final result.

Using Math::Pari, factorial( 19937 ) takes a second or so:

[ 1:32:28.64] P:\test>perl -MMath::Pari=factorial -wle"print factorial
+( 19937 )"
2.174930234150431374566653166E77066

[ 1:32:30.58] P:\test>
[download]

Using Math::BigInt takes over 100 seconds to do the same thing:

[ 1:35:13.75] P:\test>perl -MMath::BigInt -wle"$n=Math::BigInt->new( 1
+9937 ); print $n->bfac;"
2174930234150431374566653562545349453159548764005915598228287343243610
0399250094165681781366521424136575314821986705545783026313111819773987
2047105789330953704693263375025834056482446086707947547228857444817043

...
<< 1000 similar lines snipped.>>
...
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000

[ 1:37:02.13] P:\test>
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