Assumes @output is populated from your source; prints to console:

for (0..9) {
    print $output[rand($#output)] . "\n";
}
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• Printing to your final_output.txt file is left as an exercise for the student (since you've already solved that).
• This is likely to select duplicate_lines to your final file, which is still permitted by your spec (but which -- I think -- is not actually your intent). Search the Monastery for unique or @seen and similar for the many solutions to avoiding duplication.
• There is not need to create an output file (thereby eating cycles and disk space); do your initial selection from the source to an array @output.