The COUNTIFS function is designed to count things based on more than one criteria. In this case, the trick is to build a summary table first that contains one set of criteria in the left-most column, and a second set of criteria as column headings.
Then, inside COUNTIFS function, range1 is the named range "dept" (C5:C11) and the criteria comes from column F, input as the mixed reference $F5 (to lock the column). Range2 is the named range "class" (D5:D11) and the criteria comes from row 4, input as the mixed reference G$4 (to lock the row):
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