There are two ways to install support for Adobe Flash in you’re Fedora 10 (x86-64) system.You can install the i386 version of the release Adobe Flash Player (latest is flash-plugin.i386 0:10.0.15.3-release) or the alpha release of the Adobe Flash Player in x86-64 (latest is libflashplayer-10.0.d21.1.linux-x86_64.so.tar.gz)
Here I will provide the solution for both versions.
One using a yum repos channel to the Adobe website, or by downloading directly the .rpm file.
1) Preparing the Mozilla Plugins & needed libraries
- mkdir -p /usr/lib/mozilla/plugins
- yum install nspluginwrapper.{i386,x86_64} pulseaudio-lib.i386
2a) Using the yum adobe-linux-i386.repo channel
- sudo –
- rpm -ivh http://linuxdownload.adobe.com/adobe-release/adobe-release-i386-1.0-1.noarch.rpm
- rpm –import /etc/pki/rpm-gpg/RPM-GPG-KEY-adobe-linux
- yum install flash-plugin
2b) Using the flash-plugin-10.0.15.3-release.i386.rpm without the adobe yum channel
http://fpdownload.macromedia.com/get/flashplayer/current/flash-plugin-10.0.15.3-release.i386.rpm
3) Mozilla Plugin Config & Restart
- mozilla-plugin-config -i -g -v
restart firefox and enjoy 
tip: if you experience any problems with the sound, install alsa-plugins-pulseaudio, restart firefox and try again:
- yum install alsa-plugins-pulseaudio.i386
let me know how it goes…
http://labs.adobe.com/technologies/flashplayer10/
http://download.macromedia.com/pub/labs/flashplayer10/libflashplayer-10.0.d21.1.linux-x86_64.so.tar.gz