I assume that you want all tuples in which each column contains the corresponding column from either A or B. Here is one way to do it:

use strict;
use warnings;
my @A = qw(1 10 100 1000);
my @B = qw(2 20 200 2000);

my $sum = 0;
foreach my $fac1 ($A[0], $B[0]) {
    foreach my $fac2 ($A[1], $B[1]) {
        foreach my $fac3 ($A[2], $B[2]) {
            foreach my $fac4 ($A[3], $B[3]) {
                my $prod = $fac1 * $fac2 * $fac3 * $fac4;
                $sum += $prod;
                printf "%5d %5d %5d %5d: %.0f\n", $fac1, $fac2, $fac3,
+ $fac4, $prod;
            }
        }
    }
}
print $sum;
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Update: Corrected computation of products. Added print for debugging.