Yes, the amended return count as an e-filed return.
Federal returns: Although you can paper-file as many federal returns as you want using the Desktop software, only five (5) of those can be e-filed per IRS regulations.
State returns: If you have more than one state attached to your federal return, up to three (3) of them can be e-filed. Keep in mind that many nonresident state returns can't be e-filed anyway, so if you're doing more than one state, focus on e-filing the resident and part-year resident return(s).
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