If you just want an approximation, then performance may be better still if you use the fact that from Stirling's formula the log of n! is approximately

log(n**n * exp(-n) * sqrt(2*PI*n))
  = n*log(n) - n + (log(2 + log(PI) + log(n))/2
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Plug that into the fact that n choose m is n!/(m!*(n-m)!) and you can come up with a good approximation that uses Perl's native arithmetic.

This approximation may well turn out to be the fastest approach for large numbers.

Update: From Mathworld I just found out that a significantly better approximation of n! is sqrt(2*n*PI + 1/3)*n**n/exp(n). Algebraically this is less convenient, but the improved accuracy may matter for whatever you're trying to do.