If you just want a reasonable approximation (and not all the decimal places) you could use Stirling's approximation:
n! ~ (n^n)e^{-n}(\sqrt{(2)(pi)(n)})
This is an asymptotic formula (i.e. the ratio of n! to the Stirling approximation tends to 1 as n tends to infinity - it isn't true that n! is actually close to this approximation) but used on ratios of factorials of the type you need it actually gives good approximations of the values. That's certainly what I'd do if I were writing such code.
(Even if you could get an exact value, do you realy need 173 decimal places, or whatever...in many cases I would guess 2 decimal places might be pretty reasonable...for a probability.)
chas
(Update: Of course, in cases where the calculation doesn't really involve a very large number of multiplications, I would do the arithmetic directly.)