It appears that their 780 in fact does support audio through HDMI, so the problem most likely is that the software isn't set to use HDMI as default.
To fix this problem, navigate to
Control Panel > Hardware and Sound > Sound [This will bring up a pop up window] then find something like "Digital Display audio" or "HDMI Audio out" or something like that. Right click and select "Set as Default Device", and you're done!
If an HDMI audio device doesn't appear to be there, try right clicking in a blank space in the audio device window and selecting "Show Disabled Devices", then find it and set it as default.
If it still isn't there, it is possible that the card isn't detecting an audio capable HDMI device, try going back to where you set it as default and selecting "Show Disconnected Devices". If you can see an HDMI audio device then, then the problem is either
A. Your graphics card/The HDMI out port,
B. The HDMI cable,
or C, the Monitor's speakers/connection.
If this solved please don't forget to set as best answer! Thanks!