What is incumbent upon one who broke their fast for many days in Ramadan due to their inability to study while fasting: is it making up the days (qada'), expiation (kaffarah), or both?

Fasting Ramadan is one of the pillars of Islam, and breaking the fast during it is among the major sins. It is not permissible to break the fast for an examination, as it is not a religiously valid excuse. Those responsible for examinations should consider the students by scheduling them outside of Ramadan.

Regarding making up the fasts: 1. If the fast was broken due to believing it was permissible because of the examination, then you must make it up. 2. If the fast was broken with knowledge of its prohibition, then repentance is obligatory. If you broke your fast during the day, you must make it up. However, if you did not fast at all from the beginning, there is no make-up fast required for you; sincere repentance and an abundance of good deeds will suffice. 3. If there has been a delay in making up the fasts for years, you must repent for the delay. You are not obligated to pay expiation according to the most preponderant opinion, but paying it is better as a precaution.