Or he may fast by number rather than by the crescent month; in that case he must fast thirty days, for if one chooses the numerical approach it has to be thirty, and he may spread those days out.

Question: So he is free to choose either option?

Answer: Yes, he is free to choose. He can fast a lunar month, in which case he is bound by that month, or he can fast by number, in which case he must complete thirty days, whether he observes them consecutively or separately.
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Question: I made a vow to fast ten days of the month of Muharram, but because of my studies I was unable to fast during Muharram. May I fast them in another month—after my studies are over—and, in fact, I have already done so. Does that count or not? Please advise, and may Allah reward you.

Answer: If you intended the Muharram of a particular year, the opportunity has passed with the passing of that month, and you are therefore required to offer the expiation for breaking an oath, because the vow’s appointed time has lapsed. If, however, you meant an unspecified Muharram—meaning any year—then if you did not fast during this year, fast in the Muharram of another year. I consider that possibility remote, yet the question does allow for it.

As for the fasts you performed after the days had already gone by, I do not view them as a valid makeup. What is incumbent upon you is the expiation for breaking an oath because the appointed time has passed.