if 9 actors must sit together how many ways are there to seat 13 people around the table?

Since the problem specifically stated that the people gather around the table, therefore we assume a circular table hence a circular formation of gathering.  Since the arrangement is important therefore we use the permutation.  

nPr = n! / (n – r)!

However before we proceed with our calculation we MUST first affix a reference point since the formation is circle. Imagine that the reference point is the 1st point made when you draw a circle, so it is the both the 0 degrees and the 360 degrees.

Let us say that one of the actors is the reference point. So there are now only 8 actors which places can be interchanged. Since all actors have to sit together, hence the arrangements are:

8P8 = 8! / (8 – 8)! = 40,320

Now since there are 13 total people, therefore the arrangements of the other 4 people are:

4P4 = 4! / (4 – 4)! = 24

We multiply all to get the total number of arrangements:

Total arrangements = 40320 * 24 * 1                        (1 is the reference point)

Total arrangements = 967,680

15 units use socratic

step-by-step explanation: 15 units