If data is in Excel, you could use the COUNTIFS function in Excel:
=COUNTIFS(holidays.B$2:B$29;">=" & C2;holidays.B$2:B$29;"<=" & D2;holidays.D$2:D$29;">=2";holidays.D$2:D$29;"<=6")
The formula excludes weekends. You need another calculation for counting Sundays and Saturdays.
See more in attached spreadsheet.
I just looked at your question history and like to commend you for making a wise move toward asking smaller questions which eventually will enable you to complete your bigger task. Although the complete task is well described in Employee Utilization Report, I think it is a too big task for volunteer helpers, at least that was my thought when I read it first time and therefore didn't touch it.
Said differently, If you don't know how to count weekends for a period in Excel, I recommend you to make another small question for that. And after that the next small question and so on. I used this "small question strategy" when I needed to know how to make a Chain-ladder table. Because of its complexity, I didn't expect to get an answer anyway, but surprisingly did. I don't think I would if I added all issues in one question.