That is much clearer. You want something like this?

#! perl -slw
use strict;
use Data::Dump qw[ pp ];
use Algorithm::Combinatorics qw[ combinations ];


our $N //= 4;
our $K //= 2;
our $M //= 3;

my @AoA = map{
    [ map int( rand 2 ), 1 .. $M ]
} 1 .. $N;

pp 'Input:', \@AoA;

my @newAoA;

my $i = combinations( \@AoA, $K );
while( my $comb = $i->next ) {
    my @sums;
    for my $a ( @{ $comb } ) {
        $sums[ $_ ] += $a->[ $_ ] for 0 .. $#{ $a };
    }
    push @newAoA, \@sums;
}

pp 'Output:', \@newAoA;

__END__
c:\test>940693
("Input:", [[0, 1, 1], [0, 1, 1], [1, 0, 1], [0, 1, 0]])

(
  "Output:",
  [[0, 2, 2], [1, 1, 2], [0, 2, 1], [1, 1, 2], [0, 2, 1], [1, 1, 1]],
)


c:\test>940693 -N=5 -M=2 -K=3
("Input:", [[0, 1], [1, 0], [1, 1], [0, 1], [0, 0]])

(
  "Output:",
  [
    [2, 2],
    [1, 2],
    [1, 1],
    [1, 3],
    [1, 2],
    [0, 2],
    [2, 2],
    [2, 1],
    [1, 1],
    [1, 2],
  ],
)


c:\test>940693 -N=5 -M=4 -K=3
(
  "Input:",
  [
    [1, 1, 1, 1],
    [0, 1, 1, 0],
    [1, 1, 0, 1],
    [1, 1, 1, 1],
    [0, 1, 1, 1],
  ],
)

(
  "Output:",
  [
    [2, 3, 2, 2],
    [2, 3, 3, 2],
    [1, 3, 3, 2],
    [3, 3, 2, 3],
    [2, 3, 2, 3],
    [2, 3, 3, 3],
    [2, 3, 2, 2],
    [1, 3, 2, 2],
    [1, 3, 3, 2],
    [2, 3, 2, 3],
  ],
)
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