Count Distinct Values with Multiple Criteria

I have a large dataset with many employee names. I am trying to count the distinct dates based on employee for a specific date range. Some names will have many entries for the same date, so I want to know the number of days within a month that have entries. I can manipulate this with a pivot table but I need it to be with a formula. In the spreadsheet you will see I entered the results I want for each employee in D through H.

Please try at F3

=COUNT(1/FREQUENCY(IF(($B$2:$B$23>=$D3)*($B$2:$B$23<=$E3)*($A$2:$A$23=F$2),$B$2:$B$23),$B$2:$B$23))

confirm by ctrl+shift+enter