디딤돌 3 10과, the order와 the ordering 차이

“When the number was put on the calculator, the order of the numbers was kept the same.”와 “When telephones first used push buttons, the ordering of the old telephone dial was used for ordering the buttons.”라는 문장이 있어요.
the order와 the ordering은 뜻과 쓰임이 어떻게 다른가요?
주어로 쓰였을 때 ing 유무의 차이도 궁금합니다.