Answer:
24, 414,720 or 5,391,360 depending on the level of precision you (or your teacher) want. :-)
Step-by-step explanation:
The best way to approach this problem is to do it in two phases. First, consider the groups (and the top level permutations), then you consider the individuals (within the groups). There's even a third level to consider, the starting position around the table.
Before going into the solution, let's do a little warm-up using the producers group of 3. Since we talk about arranging the 13 people at the table (and not just the groups), the order of the individuals within a group is also important. Just looking at the producers, how many ways can we sit them (among themselves)? 6 ways: {a,b,c} {a,c,b} {b,a,c} {b,c,a} {c,a,b} {c,b,a}
That is governed by the following formula, since we are taking ALL the possible elements (unlike a lottery draw for example): P(n) = n!
This solution, we'll need the following grouping possibilities:
P(1) = 1, P(3) = 6, P(4) = 24 and P(5)=120
Real solution starts here
First level
First, we have to see how many arrangements are possible based on the groups...how many ways can we sit the actors, the actresses, the producers and the director as groups? There are 4 groups to consider, so the result for the groups arrangements is P(4) = 24 ways.
Second level
Now, for each of those 24 ways to arrange the groups, so they are sat together around the table) we also have to take into account the internal arrangements within each group... because as we've seen, there are 6 ways to sit the producers.
So, for the producers, we have P(3) = 6 as we've seen
For the actresses, we have P(4) = 24
For the actors, we have P(5) = 120
For the director group, we have P(1) = 1
So, for each of the 24 ways to arrange the sequence of groups, we also have 17,280 (6 * 24 * 120 * 1) ways to arrange people within the groups.
That makes a total of 24 * 17,280 = 414 720 ways to arrange the people around the table.... not taking into account the seat number.
Third level
If you want to take into account the fact that one of the 414 720 arrangements is different if the first person sits on the chair #1 than if it sits on chair #2 and so on... then we have to multiply these 414 720 arrangements by 13 to represent the various possible starting point for the arrangement. For a grand total of 5 391 360
So, the answer is your choice, 24, 414,720 or 5,391,360 depending on the level of precision you (or your teacher) want. :-)
