Hi Yetta4899,
Assuming you are covered by NFPA 70, the code is mute on specific hole size rules.
Since the cables passing through the stud are bundled for less than 24", you usually don't have to de-rate. (See next paragraph)
But, 334.80 tells us that if 2 or more NM cables carrying 2 or more current carrying conductors are passed through the same hole in the stud without spacing and the hole is sealed with thermal insulation, caulk or foam, you have two adjust ampacity based on 310.15(B)(3)(a). (This is paraphrased. I can't remember the exact code wording off of my head.)
Remember that the neutral counts as current carrying in single phase installations and to use the derating calculations based on the ampacity of the conductors, not the maximum over-current device size. (Example- #14 NM we will derate from 20 amps even through the maximum over-current allowed is 15 amps.)
More Importantly, ensure that the cables have plenty of room to expand and contract without rubbing tightly against the wood. This will, in time, abrade the jacket.
Drilling Hints-
Drilling a perfectly level hole will make pulling easier than if you angled the bit.
Laying the cables flat on top of one another will make for an easier pull with less binding & rubbing in the hole.
And, if you're using spade (paddle) bits, drilling two holes at 3/4" will be seemingly 1,000 times faster than one hole at one inch. Why? Beats me. Maybe because a 1" hole has over 1-3/4 times the area of wood to remove versus 3/4? (Auger bits don't care.)
For a clean hole with no splaying, drill from one side until the tip pokes through the other side. Then finish from the other side.
Happy Wiring!
Paul