Ok I'm just using:
We'll simplify this to
where Pr(X) is the probability the 5 people are born in different months and the set represents all possible outcomes.
Perhaps the chair analogy wasn't a very clear one. Let's go back to the months. We need to find the number of ways five people can be born on different months.
There are 12 different months and 5 people, and this is sampling without replacement, so we use permutation:
And the total number of possibilities for the months the 5 people are born in are given by because each person has 12 months to choose from. It's just the multiplication principle for counting.