Question 2: remember that there are 5 lead male roles and 4 lead female ones. if you must seat the male leads together and the female leads together, the three producers together and the director by himself, how many different ways can you seat 13 people around the circular table?

 103,680


Explanation:

There are 13 people total. We can group up the male leads since they must sit together. Consider this group of five as one "person". Do the same for the female leads since they must sit together as well. The same applies to the producers.

Effectively we have these groups
* Male leads
* Female leads
* Producers
* Director

So we have 4 "people" in a sense sitting around the table giving (n-1)! = (4-1)! = 3! = 3*2*1 = 6 different ways to arrange these four groups.

Within the "male leads" group, there are 5! = 120 ways to permute the members.
Within the "female leads" group, there are 4! = 24 ways to permute the members.
Within the "producers" group, there are 3! = 6 ways to permute the members.
Within the "director" group, there's only one person so 1! = 1 way to arrange the members
So there are 120*24*6*1 = 17,280 different ways to arrange any one of the six permutations of the four groups.

Overall, there are 6*17,280 = 103,680 total permutations.