I stand by my (quite conservative) statement. It does introduce errors for fairly small values of n. It is also at least twice as slow (it can be orders of magnitude slower).

But the greatest weakness to my mind is the following output:

200 choose 1: 200 vs. -1.#IND (1.#QNAN).
        Rate  easy  nice
easy  1464/s    --  -96%
nice 34687/s 2270%    --
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Yes, there is a useful space over which its accuracy is quite good (though not always as good), and thanks for exploring that.

Here is the code I used:

#!/usr/bin/perl -w
use strict;
use Benchmark qw( cmpthese );
sub fact {
  my $fact = 1;
  $fact *= $_ for 1..$_[0];
  return $fact;
}
sub nice {
  my ($n, $r) = @_;
  my $res = 1;
  for my $i (1..$r) {
    $res *= $n--;
    $res /= $i;
  }
  return $res;
}
while(  <>  ) {
    my( $n, $m )= split ' ';
    my $nice= nice($n,$m);
    my $easy= fact($n)/fact($m)/fact($n-$m);
    print "$n choose $m: $nice vs. $easy (",abs($nice-$easy),").\n";
    cmpthese( -3, {
        nice=>sub{nice($n,$m)},
        easy=>sub{fact($n)/fact($m)/fact($n-$m)},
    } );
}
[download]