Question

How many ways can 8 people be seated around a circular table if

(a)There are 4 men and 4 women, and no 2 men or 2 women can sit next to one another?

(b)There are 5 men and they must sit next to each other?

(c)There are 4 married couples and each couple must sit together?

Answer 1

Let us consider the position of Men and then the Women.

To meet the condition that no two men can sit next to each other, this is the possible way in which the men can sit.

M * M * M * M *

(Here, * represents the vacant place)

Now, these men can be seated by 4! ways.

Now the remaining vacant places will be filled by women and they can be seated in 4! ways.

So the total is 4!*4!

The will be multiplied by 2 because, in the seating arrangement the first person to sit can be Woman.

W * W * W * W *

So the final result will be 4! * 4! *2

4! (The order of males)× 4! (The order of females)× 2 (the first person is either male or female) = 24×24×2 = 1152

First arrange the 4 men around circular table in alternate chairs in (4 - 1)! = 3! ways. Now four alternate chairs are vacant.

Each of the 4 women can now be seated in the vacant chairs in 4! ways.

Total number of ways = 4! X 3! = 144 ways.

Why number of ways for men is 3! and not 4! ?

It is because it is immaterial where the first man is seated. The problem of uniqueness/permutation begins only after the first man is seated. Therefore, only three men have to be seated in some order now and they can be seated in remaining 3 seats in 3! ways !!!

All 4 women have 4 relative positions to occupy