my $nregions = 354;

sub random_subset {
  my ($n, $k) = @_; # $k member subset of 1..$n.
  my %member;
  while ($k > 0) {
    my $x = int(rand()*$n)+1; # generates random number 1..$n.
    redo if $member{$x};
    $member{$x} = 1;  
  } continue {
    $k--;
  }
  sort keys %member;
}

sub insert_subset {
  my ($dbh, $generation, @members) = @_;
  my $sth = $dbh->prepare("INSERT INTO Random_region_lookup_table_TEMP (Generation_number, Place_key) VALUES (?,?)");
  for my $x (@members) {
    $sth->execute($generation, $x);
  }
}

...
my $dbh = DBI->connect(...); # fill in your connection parameters here
...
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($dbh, $generation, @members);
}