Question

Permutations!!

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

Answer 1

To calculate the number of ways to seat 13 people around a table with 9 actors sitting together, we treat the 9 actors as one unit and then arrange the five units around the table, resulting in (4! * 9!) different possible arrangements.

Step-by-step explanation:

The question asks us to calculate the number of ways to seat 13 people around a table if 9 actors must sit together. This can be approached as a permutations problem in combinatorics.

Firstly, treat the 9 actors as one unit since they must sit together. With this in mind, we effectively have 5 units to arrange: the group of 9 actors and the remaining 4 individuals. As the seating arrangement is around a circular table, we can fix one person's seat and arrange the remaining units. As a result, there are (5-1)! ways to arrange these units since circular permutations eliminate the concept of a distinct 'starting' point that linear permutations have.

Now we need to consider the arrangements of the 9 actors within their group. Since their relative positions to each other matter, they can be permuted in 9! ways.

Therefore, the total number of seating arrangements would be the product of the two permutations: (5-1)! * 9!.

Calculating this gives us (4!) * 9! = (4*3*2*1) * (9*8*7*6*5*4*3*2*1) different possible arrangements.

Answer 2

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