Any day is ok for the first student.
Any day except the first student s b-day is ok for the second, any but those of the 1st and 2cd are ok for the 3d, etc. In the course you will learn general formulas for how many ways you can put N objects in M bins, depending on whether multiple occupancy is allowed and on whether the objects are distinguishable. This problem amounts to asking how many ways you can put 30 distinguishable students in 365 bins without allowing multiple occupancy, compared to how many ways you can put them in 365 bins with multiple occupancy allowed.