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use Math::Combinatorics qw( permute );

...
   $s =~ s/$others//g;
   return $valid{$s};
}

use Math::Combinatorics qw( permute );

...
   $s =~ s/$others//g;
   return $s =~ $valid;
}

use Math::Combinatorics qw( permute );
use Regexp::List qw( );
...
   $s =~ s/$others//g;
   return $s =~ $valid;
}