As we discussed in that meeting, here's the code snippet I was thinking about to generate the binary strings:

print map "$_\n", strings_for(6, 4);

sub strings_for {
  my ($cats, $cols) = @_;
  $cats--; $cols--;
  my @ret;
  for (0..(1 << $cats) - 1) {
    my $bitstring = substr(unpack("B*", pack "N", $_), -$cats);
    next unless $bitstring =~ tr/1// == $cols;
    push @ret, $bitstring;
  }
  @ret;
}
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-- Randal L. Schwartz, Perl hacker