If I can suggest a follow up post Rick -- I would be interested in a simulation approach to figure out the probability just limiting to school days. So you have birthdays not on school days, and fewer days to fill in. So not clear to me how the solution holds exactly for the more restricted question "should there be a birthday on everyday we are in school".

Also given the large number of permutations and low probabilities, figured you would have some good insight to share on simulating a solution, checking its accuracy, now many iterations needed etc. (assuming a closed form formula is not possible).