Count a specific date within a range of multiple dates

Hi,
I'm struggling with working the code that will enable me to count the number of times a specific date falls between ranges of multiple dates. I'm trying to automate a system to allow me to see how many people are at work on any given day of the month by knowing when they are on rotation.
E.g.
Mr M Smith is at work 22nd Oct 2015 until 20th Nov 2015 so need to show as a number that he is at work on 23rd Oct 2015.
Miss S White is at work 29th Oct 2015 until 30th Nov 2015 so need to show as a number if she is at work on 28th Oct 2015
Etc.
I've attached the spreadsheet I'm working with. I've only been able to show "FALSE" or "YES"/"NO" for the adjacent dates not the whole range in columns C and D. I’d like the result to show a number i.e. 10 or 20 people are at work on 28th Oct 2015 etc.

Any help would be much appreciated.
Thanks