sub insert_subset {
  my ($sth, $generation, @members) = @_;
  for my $x (@members) {
    $sth->execute($generation, $x);
  }
}

my $dbh = DBI->connect(...); # fill in your connection parameters here
my $sth = $dbh->prepare("INSERT INTO Random_region_lookup_table_TEMP (Generation_number, Place_key) VALUES (?,?)");
...
my %seen;
for my $generation (1..10) {
  my @members = random_subset($nregions, 5); # 5 element subsets, e.g.
  my $key = join(" ", @members); # note: members already sorted
  redo if $seen{$key}; # make sure the subsets are unique
  $seen{$key} = 1;
  insert_subset($sth, $generation, @members);
}