I'd not store the index, where the file begines, but the number of words in it.

Searching is then done by substracting the number of words from the random number.

As a final step, the file found is swapped with the last file and the number of words is substracted.

#!/usr/bin/perl
use strict;
use warnings;
my @number_of_words = (
    { number => 10, name => 'file_a.txt' },
    { number =>  1, name => 'file_b.txt' },
    { number =>  3, name => 'file_c.txt' },
    { number =>  5, name => 'file_d.txt' },
    { number => 10, name => 'file_e.txt' },
);

my $total_number_of_words = 29;

my $select= 3;

my $selected= 0;

while ( $select-- ) {
    my $randomindex= rand $total_number_of_words;
    my $last_index= ( scalar @number_of_words ) - $selected - 1;
    my $i= $last_index+1;
    do {
        $randomindex-= $number_of_words[--$i]->{'number'};
    } while ( $randomindex >= 0 );
    $total_number_of_words-= $number_of_words[$i]->{'number'};
    # print $number_of_words[$i]->{'name'},"\n";
    @number_of_words[$i, $last_index]= @number_of_words[$last_index, $
+i];
    ++$selected;
}

for (my $i= $#number_of_words - $selected; ++$i<=$#number_of_words;) {
    print $number_of_words[$i]->{'name'},"\n";
}
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