You can carry as many as you like. It doesn't matter as they count on strokes not balls. However, you are expected to use the same ball from tee to the hole unless you loose it, or it becomes unplayable due to damage. You can play a new ball on every hole if you wish.
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A tour pro would have around a dozen golf balls in there bag per round, just in case. But it is completely up to the player.
You would find that a tour pro would use between 12 and 20 golf balls for a four round tournament.
Wilson sends 80 thousand balls to the US Open every year. Only about 70 thousand of them are used. The rest are givin to local court communitee programs