No, there are 3 conductors but 4 total wires (the AC ground is the 4th). The neutral in Alternating Current wiring is an active "conductor" and the "ground" is a safety device. That's an unfortunate terminology difference from Direct Current wiring, where the "ground " is a conductor (the negaive terminal wire path). So in wire nomenclature, 10/3 describes a cable with 3 conductors, at least 2 of which are 10 gauge wire (the neutral is sometimes a lesser gauge). Most such cables also have a 4th wire and will state "with ground", but it's still 10/3 wire whether it includes the ground or not. Confusing to an amateur, eh?10/3 and 6/3 is two conductors and a ground. Three wires.
So the question is, can I use the wire I have or do I need to go buy 6/3 and 10/3? Or run the x/2 wires and put "bare copper" where green is shown in the diagram.No, there are 3 conductors but 4 total wires (the AC ground is the 4th). The neutral in Alternating Current wiring is an active "conductor" and the "ground" is a safety device. That's an unfortunate terminology difference from Direct Current wiring, where the "ground " is a conductor (the negaive terminal wire path). So in wire nomenclature, 10/3 describes a cable with 3 conductors, at least 2 of which are 10 gauge wire (the neutral is sometimes a lesser gauge). Most such cables also have a 4th wire and will state "with ground", but it's still 10/3 wire whether it includes the ground or not. Confusing to an amateur, eh?
You don't have to combine all the necessary wires in one cable (wrapped in one external covering), nor do you have to use bare vs insulated wire for grounds. Green insulated wire is fine, and so is bare copper. Yes, you can connect bare copper ground to a green-insulated ground. Since you have the 10/2 and 6/2 cable, both of which have a ground wire in the bundle, you can use them.So the question is, can I use the wire I have or do I need to go buy 6/3 and 10/3? Or run the x/2 wires and put "bare copper" where green is shown in the diagram.