From Fedora Project Wiki

Enabling Flash Plugin

For i386

Unlike earlier releases, no special steps are required to install the proprietary Adobe Flash Player 10 on Fedora 10.

For X86_64

Select the YUM version of the Adobe Flash Player from Adobe's download page. This installed Adobe's software repository so that updates can be applied by the normal Fedora update process.

Next some helper files are needed in addition to the Flash Player: 

su -c 'yum install flash-plugin nspluginwrapper.i386 \
    alsa-plugins-pulseaudio.i386'
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