He vowed to fast for ten days if he passed his exams; does he have to fast them consecutively?

Firstly:

If someone vowed to fast for ten days if he passed his exams, he must fulfil his vow if he attained his goal, because this comes under the heading of a vow to do an act of worship, and a vow to do an act of worship must be fulfilled. The Prophet (blessings and peace of Allah be upon him) said: “Whoever vows to do righteous deed, let him do that righteous deed, and whoever vows to do a sinful deed, let him not do that sinful deed.” Narrated by al-Bukhari (6202).

It says in al-Insaf (11/130) by al-Mardawi (may Allah have mercy on him): Once the condition is met, then his vow becomes binding and he must fulfil it, and there is no difference of scholarly opinion regarding that. End quote.

Secondly:

Whoever vows to fast for ten days, but does not specify or intend that they should be consecutive, may fast them consecutively or separately.

Al-Bahuti (may Allah have mercy on him) said: If he vows to fast a certain number of days, even thirty days, he does not have to do them consecutively, because mentioning days does not indicate that they are consecutive, based on the verse in which Allah, may He be Exalted, says (interpretation of the meaning): {then an equal number of days [are to be made up]} [al-Baqarah 2:184]. That applies unless he stipulated or intended that they should be consecutive, in which case he must fulfil his vow.

End quote from Kashshaf al-Qina`, 6/282.

In Nihayat al-Muhtaj (10/282) it says: If he stipulated whether they should be done separately or consecutively, then what he stipulated becomes binding and he must do it as he committed himself to do it. In the case of consecutive fasts, the matter is clear. In the case of doing the fasts separately, this is comparable to the fasting of the pilgrim who is doing hajj tamattu`: if he vows to fast ten days separately, then he fasts them one after another, only five of them will count for him. But if he did not specify whether he would fast the days separately or consecutively, he may do either, but doing them consecutively is better. End quote.

Shaykh Ibn `Uthaymin (may Allah have mercy on him) was asked:

My mother became very sick, and she vowed that if Allah healed her from this sickness, she would fast for nine days every month. Then she recovered from her sickness by Allah’s leave, and she fasted these days for several months, nine days every month. But then because of some life circumstances, such as having to tend the sheep, sow the crops, harvest the crops and so on, she was no longer able to fast these days, so she limited it to three days every month. Until now, she has not stopped fasting these three days every month. What must she do for not fasting the remaining six days of her vow? What must she do if she wants to free herself from this vow?

He replied: If a person vows to do an act of worship, whether the vow is without conditions or there is a condition attached to it, then he must fulfil that vow and do that act of worship which has become obligatory for him, because the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever vows to do righteous deed, let him do that righteous deed.” Your mother made a vow that she can fulfil, which is to fast nine days of every month, if Allah, may He be Exalted, healed her. Allah indeed healed her from her sickness, and this vow is connected to healing from it and she is able to fast these nine days spread throughout the month. So she can fast three days in the first ten days, three days in the middle ten days, and three days in the last ten days of the month, so long as she did not stipulate or intend that they should be consecutive. She cannot free herself from that, unless she becomes unable to do it in a way that makes it permissible for her not to fast in Ramadan, such as sickness and the like. In that case, she may stop doing that because of this excuse, and she can make it up later on.

End quote from Fatawa Nur `ala ad-Darb.

Based on that, if this student did not stipulate or intend to fast these days consecutively, then he does not have to do that. But if he did stipulate or intend that, then he must do it.

And Allah knows best.