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 )"
[ 1:32:30.58] P:\test>
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;"
<< 1000 similar lines snipped.>>
[ 1:37:02.13] P:\test>
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