This is a general code critique.

# Always use strict especially when you are new. Its a boon to debuggi
+ng and it helps you from screwing up in some interesting ways (but mo
+stly it does some typo-checking)
use strict;
use warnings;

open (TEXTFILE, 'text.txt') or die ("Cannot open file : $!");

# $/ is normally "\n" which is what readline() uses to decide when to 
+stop reading. You want to assign undef here so that readline() will p
+ull in the entire file and not just the first line of the file.
$/ = undef;

# strict will force you to declare your variables. Here I use my() to 
+make $big_string be a lexical.
my $big_string = <TEXTFILE>;
 
$big_string = lc ($big_string);

# I re-wrote this as a tr/// expression so it would be so ugly as a re
+gular expression. If you really wanted to use s/// then s/[()]//g wou
+ld have been preferred.
$big_string =~ tr/()//d;

# I declared @words with my() and moved the plus sign out of the chara
+cter class. You meant to split on repeated white-space but you actual
+ly just indicated that plus signs should also be split on. Moving the
+ + quantifier onto the character class indicates that any repeated ma
+tching of the class should be smushed together.
my @words = split (/[,.\s]+/, $big_string);

# I replaced your for/push loop with a grep which only passes words th
+at are three characters or longer. You should also note that in your 
+original expression that anything that matches {4,} is also going to 
+match {3,}. If you really meant to do that then just the single {3,} 
+would have been good enough. Also, instead of saying [a-zA-Z] you cou
+ld have written /[a-z]{3,}/i which is case insensitive. 
my @gt_three_char_words = grep /[a-z]{3,}/i, @words;

# Rename %hash to %word_counter and declare it.
my %word_counter;
foreach (@gt_three_char_words) {
    $word_counter{$_}++;
}

# Sort the list of words so you can then extract whatever portion you 
+like.
@gt_three_char_words = sort {
        # Sort the word list by occurrances
        ( $word_counter{$a} <=> $word_counter{$b} ) ||
        # Then alphabetically
        ( lc( $a ) cmp lc( $b )
    } @gt_three_char_words

# Print the 
print
    join( '',
          map( "$_: $word_count{$_}\n",
               @gt_three_char_words[ 0 .. 9 ] ) )
    . "\n";
[download]

---

Now here is how I would have written the entire thing.

use strict;
use warnings;
use Getopt::Long;
use File::Slurp;

my $filename = 'text.txt';
my $min_word_length = 3;
my $start_word = 0;
my $words_to_display = 10;
GetOptions(
    "file=s" => \$filename,
    "min=i" => \$min_word_length,
    "start=i" => \$start_word,
    "words=i" => \$words_to_display
);

my $original_text = lc read_file( $filename );
$original_text =~ tr/()//d;

my @words =
    grep /[a-z]{$min_word_length,}/i,
    split /[,.\s]+/,
    $original_text;

my %word_counter;
$word_counter{ $_ } ++ for @words;

@words = sort {
        $word_counter{ $a } <=> $word_counter{ $b } ||
        $a cmp $b
    } @ words;

my @words_to_print =
    @words[ $start_word ..
            ( $start_word + $words_to_display - 1 ) ];

for my $word ( @words_to_print ) {
    print "$word: $word_counter{ $word }\n";
}
[download]

---

---

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