Usually, the play that the team is using determines which player inbounds the ball. (Either after a foul, out of bounds, or shot.) Most of the time one of the guards/wings will inbound the ball with the point guard, but sometimes the person who catches the ball after the basket will just step out of bounds and pass it in.
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No position always takes out the ball. Most likely, a center or power forward will take out the ball and pass it in to the point guard. However, anyone can take out the ball.