Count if using multiple date criteria

Hello:

I am working on a spreadsheet that records inventory. I need to create a
formula that counts if any items were completed late by a person . The
information regarding the person is recorded in one of two columns (M and N).
The date completed is recorded in column O. The due date is column G. I
also need to record this for each month received. The received date is
column A. So for example, I need to know how many items received in November
2009 and assigned to J. Smith were completed after the due date.

Thanks for any help you can give!