Sometimes when you uninstall Office 2013 it doesn't remove all the components.
Office 15 Click-to-Run Extensibility Component remains on the computer and you might get an error when you try to reinstall Office 2013
To uninstall Office 2013 completely you must manually uninstall the Office 15 Click-to-Run Extensibility Component
32bit C:\Program Files\Microsoft Office 15\root\Integration\c2rint.msi 64bit C:\Program Files (x86)\Microsoft Office 15\root\Integration\c2rint.msi
Right Click on c2rint.msi and Selected Uninstall.
Now you'll be able to re-install Office 2013