How I setup Flash on my fresh Fedora 10 x86_64 install.
Close Firefox
Change to the Firefox plugins directory:
cd /usr/lib64/mozilla/plugins/
Download the Flash plugin:
wget http://download.macromedia.com/pub/labs/flashplayer10/libflashplayer-10.0.d20.7.linux-x86_64.so.tar.gz
Extract the plugin:
tar -xvzf libflashplayer-10.0.d20.7.linux-x86_64.so.tar.gz
Remove the .tar.gz as you now have the plugin extracted:
rm libflashplayer-10.0.d20.7.linux-x86_64.so.tar.gz
Open Firefox & you should be away!