In a certain university, there are three types of courses offered: Mathematics, Science, and Humanities. Each type of course has a different number of classes. The Mathematics department offers 4 classes, the Science department offers 5 classes, and the Humanities department offers 3 classes. If a student wants to enroll in one class from each department, how many different combinations of classes can the student choose? Additionally, if the student decides to drop one of the chosen classes, how many different ways can the student choose the remaining classes? Let's think step by step and output the final answer within \boxed{}.
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