Here is a simpler program that accomplishes the same thing for up to 32 checks:

my @check = (10, 5, 13, 3, 15, 1);
my $desired = 16;

foreach my $index (0..2**@check-1) {
   my $sum = 0;
   foreach my $pos (0..@check-1) {
      my $bit = ($index >> $pos) % 2;
      $sum += $bit * $check[$pos];
   }
   if ($sum == $desired) {
      my @combo;
      foreach my $pos (0..@check-1) {
         push @combo, $check[$pos] if  ($index >> $pos) % 2;
      }
      print join " ", @combo, "\n";
   }
}
[download]

Here I index the possibilities with a simple integer expressing a binary combination of checks.

Timing was 55 milliseconds on an Athlon 2100.

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