Using countifs with two variables searching a range of columns

I might be over simplifying this, but why not put your data in a table with a total row?  You can filter by any number of the columns in any combination and return the number of rows match your criteria.  Just click the arrow next to the header, uncheck the value you do not want counted, then click ok.  To reset, with any cell in the table selected, you can go to the Data tab, and click the Clear button in the Sort & Filter section, or clear each column individually, by clicking the arrow like before, and selecting Clear Filter from "<columnname>"