#!/usr/bin/env perl

use strict;
use warnings;
use Algorithm::Loops qw(NestedLoops);
use Text::Levenshtein::XS qw(distance);

my @chars = qw(A M N U);

my $depth    = 5; # We're taking "five at a time" from A, M, N, U.
my $min_dist = 3; # We need a minimum difference of 3 characters.

my @strings;

NestedLoops(
    [ ([@chars]) x $depth ],
    sub { push @strings, join('', @_) },
);

my $count = 0;
foreach my $left (@strings) {
    foreach my $right (@strings) {
        if (distance($left, $right) >= $min_dist) {
            print ++$count, ": $left <=> $right\n";
        }
    }
}

print "\n$count pairs with minimum difference of $min_dist from a set 
+of ",
      scalar(@chars), " characters taken $depth at a time.\n";
[download]

Algorithm::Loops and Text::Levenshtein make it easy to generate the list of hits.

Output:

1: AAAAA <=> AAMMM
2: AAAAA <=> AAMMN
3: AAAAA <=> AAMMU
4: AAAAA <=> AAMNM
5: AAAAA <=> AAMNN
.... # many lines deleted
917168: UUUUU <=> UUNMM
917169: UUUUU <=> UUNMN
917170: UUUUU <=> UUNNA
917171: UUUUU <=> UUNNM
917172: UUUUU <=> UUNNN

917172 pairs with minimum difference of 3 from a set of 4 characters t
+aken 5 at a time.
[download]

Great solution. Thanks a lot :)