I hope you use yum, and if you do, your problem can be solved in a matter of minutes.
First, do the following:
1. Open a Terminal.
2. Become root:
3. Now, install my yum configuration (Updated 03 Feb 2007), using this command:
rpm -Uvh http://www.fedorafaq.org/yum http://rpm.livna.org/livna-release-6.rpm
When you're done with that, your yum has been configured and is up to date.
Then do this:
# Open a Terminal.
# Become root:
yum install flash-plugin
Recommended read: fedorafaq.org