This question came about when I was with a group of 8 friends (i.e. 9 in total), and we realised we all had birthdays in distinct months - we tried to work out the probability of this happening. The 2 answers we came up with seemed either too remote or too common:
The probability that I have a birthday in any given month is $\frac{1}{12}$, and so the probability that all the other people have birthdays in a different month is $\frac{8}{11}$; adding these together and multiplying by the number of people:
$ (\frac{1}{12} + \frac{8}{11})^9 $
gives 0.15 - which seems too big (and I've realised can't be correct because it gives an impossible answer when the number of friends is >11)
The second way I thought it could be calculated was
$\frac{1}{12} * \frac{1}{11} * \frac{1}{10} * \frac{1}{9} * \frac{1}{8} * \frac{1}{7} * \frac{1}{6} * \frac{1}{5} * \frac{1}{4}$
but this gives 1.25e-8, which seems too small.
What's the correct way to calculate it?!
Speak out one by one in which month your birthday falls.
The probability that $9$ distinct months will be mentioned is:$$\frac{12}{12}\times\frac{11}{12}\times\frac{10}{12}\times\cdots\times\frac5{12}\times\frac{4}{12}=\frac{12!}{3!12^{9}}$$
E.g. the factor $\frac{10}{12}$ denotes the probability that the month mentioned by the third persons will not be a month allready mentioned under the condition that the months that were allready mentioned are two distinct ones.
The first person will mention a month. Then the probability that the second person will mention a month that differs from the first one is $\frac{11}{12}$. Then - if both mentioned months are indeed distinct - the probability that the third person will mention a month different from both is $\frac{10}{12}$. Et cetera.