More refactoring…

#!/usr/bin/perl

use strict;
use warnings;

# Get the input...
print "\nEnter the power (a positive integer from 1 through 33): ";
chomp(my $power = (<STDIN>));

# Quietly end if there's no input...
$power =~ m/\S/ or exit 1;

# Validate the input...
$power =~ m/^(?:[1-9]|[12][0-9]|3[0-3])$/
    or die "$power is not a positive integer from 1 through 33\n";

# Expand the binomial...
my $expansion = binomial_expansion(1, $power, 1);

# ...and then print the result...
print "(x + y)^$power = $expansion\n";

exit 0;

sub binomial_expansion {
    my ($k, $x, $y) = @_;

    my $term = sprintf "%dx^%dy^%d", $k, $x, ($y - 1);

    $term =~ s/\b1(?=x)//;            # Remove 1 from 1x
    $term =~ s/(?<=[xy])\^1(?!\d)//g; # Remove ^1 from x^1 and y^1
    $term =~ s/[xy]\^0//;             # Remove x^0 and y^0

    return $x > 0
         ? sprintf "%s + %s", $term,
           binomial_expansion(($k * $x) / $y, ($x - 1), ($y + 1))
         : $term
         ;
}
[download]