In how many ways can 8 people be seated in a row if there are no restrictions on the seating arrangement?

persons A and B must sit next to each other?

there are 4 men and 4 women and no 2 men or 2 women can sit next to each other?

there are 5 men and they must sit next to one another?

there are 4 married couples and each couple must sit together?

a) P₈ = 40320

b) P = 576

c) P = 1440

d) P = 24

Step-by-step explanation:

a) 8 people sitting in a row without restrictions

simple Total ways = P₈ = 8!

P₈ = 8*7*6*5*4*3*2*1

P₈ = 40320

b) There are 4 men and 4 women and no two men or 2 women can sit next to each other

Let letters e women and numbers be men we have something like this

1 a 2 b 3 c 4 d

So we have the permutations of the four digits (men)

P₄ = 4!

P₄ = 4*3*2*1 = 24

And the permutations of the 4 women too equal to 24

Then total ways of sitting in a row in this case is

P = 24*24 = 576

c) There are 5 men and they have to sit next to each other

In this case we have 2 groups:

Group 1 5 men Group 2 3 women

We have two groups and the ways are

group men first group of women second 1 way

group of women first group of men second 2 way

Permutations within men group

P₅ = 5! = 5*4*3*2*1

P₅ = 120

Permutations within the women group

P₃ = 3! = 3*2*1

P₃ = 6

Total ways case c) T = 6*120*2 =

P (c) = 1440

d) There are four couples and each couple must be together

P₄ = 4! = 4*3*2*1

P₄ = 24