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The math teacher has three classes.
This will lead to duplication, because whether the teacher assigns first or later depends only on the final result, such as your method. If teacher 1 is assigned to class 1, no matter which time, A(3/5) or the second time, it is division. This is a combination problem, not an arrangement problem.

It can be calculated that the final score is 2 21. Choose one of the three classes and assign a teacher C(3/ 1). Assign one teacher to this class, and the remaining four teachers are assigned to two classes, with two teachers in each class. If graded, it is C(3/ 1).