Suppose we have 20 people and ask their birthdays. What is the probability that 4 months have 3 birthdays and 4 months have 2 birthdays( and 4 months have no birthdays)?

Description
Suppose we have 20 people and ask their birthdays. What is the probability that
4 months have 3 birthdays and 4 months have 2 birthdays (and 4 months have no
birthdays)?
Solution
The sought probability is
n
m
p  , where n is the full number of possibilities in
which 20 people will “choose a month to be born”, and m is the number of
configurations in which 12 out of those 20 people have been evenly distributed between
4 months and the remaining 8 people have been uniformly spread over another 4
months, leaving no people for the remaining 4 months of the year.
The easy part is that the denominator is given by...