Sorry, there was an error in the formula,
Here is one, that also screens out the year:
Example for Person 1, Cat A, Month 2, Year 2009

=SUMPRODUCT((B1:B1000=1)*(C1:C1000="A")*(E1:E1000 =DATE(2009,2,1))*(E1:E1000<=DATE(2009,3,0)))

Rgds. Hansueli