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If such a conflict ever occurred, how many of each would be needed in order to take down Vader in your opinion?

E.g
It would take x number of Royal Guards in fight a, and x shadow guards in fight b.

Starting distance is what it would be in the Emperor's throne room.

4-6 Shadow Guards for lightsaber only combat; about 3 Shadow Guards and 10 Royals; unknown amount of Royal Guards.

Can't say about full combat with Force included.

The throne room isn't that big, so Vader may not be able to bring his full Force power to bare with numbers against him.

Though the numbers also limit the SGs and RGs too along with the size of the throne room, they won't be able to attack all at once against Vader either.

The RGs would have more issues than the SGs however.