Answer: 103,680
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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.
