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4 June, 08:52

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?

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  1. 4 June, 11:27
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    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
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