Question

There are 5 lead male roles and 4 lead female roles. If you must seat the male leads together, the female leads together, the 3 producers together, and the director by himself, how many different ways can you seat the 13 people around the circular table?

Answer 1

24, 414,720 or 5,391,360 depending on the level of precision you (or your teacher) want. :-)

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. :-)

Answer 2

Answer with explanation:

Number of Male lead roles = 5

Number of female lead roles =4

Number of Producers =3

There is a single director.

Total number of people who will sit around circular table

=5+4+3+1

=13

The answer can be explained in the following way.Suppose there are three people A,B and C who have to sit around a circular table.

When it has to be arranged in linear way it can be arranged in 3! ways which are: A B C,A CB, B AC,B C A,C A B,C BA.

When these are arranged in circular arrangement,ABC , B CA,CAB are same.

So,in a linear arrangement each six of the arrangement of ABC is different and there are three different Circular arrangement of ABC.

So, total number of Circular Arrangement

`=(3!)/(3)\\\\=2`

As,3!=3 ×2×1

→→So, total number of arrangement of 13 different person when they are seated around a circular table such that male leads together, the female leads together, the 3 producers together, and the director by himself

=There are four things that is 5 male, 4 female,3 Producer and 1 Director considering male as 1, female as 2, Producer as third and director as 4,there are 4 things which need to be arranged in a circular table

`\rightarrow (4!* 5! * 4! * 3! * 1!)/(4)\\\\=(24 * 120 * 24 * 6)/(4)\\\\=24 * 120 * 6 * 6\\\\=103680`

→n!=n×(n-1)×(n-2)×(n-3).........1