ok, download for free Revo Uninstaller from
www.revouninstaller.com just get the 30 day trial.
Run it and use it to remove any reference to Office 2010 and 2013.
Then restart the computer
than do the following please
Download CC Cleaner (free) from hereCC Cleaner
(Click Download Latest Version on the top right)
Run it and let it scan and repair your registry.
Once done, restart the computer
Press the Windows Key + X and click Command Prompt (admin)
A black window will open.
Type in it
press enter and when it has completed, type in
It will complain the Volume is in use and to run on restart. Say YES and restart the computer
Than install Office 2013 and tell me please if you still have the problem.