Is it permissible to shorten the prayer (qasr) in university housing that is 100 km away from one's place of residence, and what is the ruling on shortening the prayer in this case—is it obligatory or permissible—knowing that the duration of stay in the housing has reached 15 days?

A distance of 80 kilometers is considered a distance that permits shortening prayers (qasr), and it is permissible to shorten prayers during travel. If you arrive at your destination and intend to stay for 15 days, it is not permissible for you to shorten prayers; rather, you must offer them in full (itmam). This is the preponderant view among the majority of jurists for anyone who intends to stay for four days or more. Shortening prayers during travel is not obligatory except according to the Hanafis and one narration from Malik. The preponderant view is that shortening prayers is better than offering them in full.