I have refactored your code:

• no global variables are used by your function
• all formatting is done within the function
• you no longer get a trailing plus sign (+)
• added check on input

use warnings;
use strict;

# Get the input
print "\nEnter the power: ";
chomp(my $power = (<STDIN>));
if (($power =~ /\D/) or ($power == 0) or ($power > 9)) {
    die "Error: $power is not a positive integer less than 10";
}

# Do the work
my $result = binomial_power(1, $power, 1);

# And print it out
print "The result is: \n(x + y)^$power = $result\n";


sub binomial_power {
    my ($c, $xp, $yp) = @_;
    my $result = $c . "x^" . $xp . "y^" . ($yp - 1);
    $result =~ s/1x/x/g;
    $result =~ s/([xy])\^1/$1/g;
    $result =~ s/[xy]\^0//g;
    if ($xp > 0) {
        $result . " + " . binomial_power((($c * $xp)/$yp), ($xp-1), ($
+yp+1));
    }
    else {
        return $result;
    }
}
[download]

While I haven't eliminated the formatting regexes, I have combined a couple of them.

Please fix the spelling of "Binomals" in your title; it will help others find your code when searching.

UPDATE: changed input checking to reject power > 9, since our code does not support it.

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