How is it done? Do the discs come with an activation code?
If I can, do I need the physical copies of Mass Effect 3 or can I download the game through Origin from that point on?
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Physical copies come with an Origin-redeemable key inside. In fact, you'll need to register it with Origin to install the game in the first place.
If you already have the discs available, you might as well use them to save some bandwidth, but after registering the key, you'll be able to download the game from Origin as well.