blokhead's approach is certainly the most general, and I'm glad he posted it. My naive approach for this case, however, would have been to use a simple modification of the Fisher-Yates shuffling algorithm (untested and not optimized):

sub random_pick {
  my ( $n, $arr ) = @_;
  my @idx = 0..$#$arr;
  my $i = 0;
  while ( $i < $n ) {
    my $j = $i + rand( @idx - $i );
    @idx[ $i, $j ] = @idx[ $j, $i ];
    ++$i;
  }
  return @{ $arr }[ sort { $a <=> $b } @idx[ 0 .. $n-1 ] ];
}
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This samples randomly with replacement, but you can keep track of the picks (in a hash), and discard any repeats. As blokhead noted, for the numbers you're looking at this would happen very rarely, and probably never during your use of the algorithm.

Zaxo's approach also uses FY, but doing a full shuffle each time. On the one hand this is more work than is necessary, but on the other List::Util::shuffle is a compiled extension, so it may end up being faster than the solution above.