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The required size of a conduit raceway for a 3-5000 mcm wire is 300 kcmil. It maximizes the limits on the wire fill for raceways.
You can pull 10 #3 gauge wires in a 2 1/2" conduit.
For new 200 amp services the conduit size should be 3 inch PVC.
A trade size hole for a 2-1/2" EMT conduit connector, the hole needs to be 2-7/8" in diameter. A 3" rigid conduit can also use this size hole without the lock nuts falling into the hole.
For a distribution panel rated at 400 amps parallel runs of 3/0 conductors will do the job.