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Thread: Permutation Question

  1. #1
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    Permutation Question

    Question:

    Six natives and two foreigners are seated in a compartment of a railway carriage with four seats either side. In how many ways can the passengers seat themselves if
    a) the foreigners do not sit opposite each other,
    b) the foreigners do not sit next to each other?

    a) 6 natives 2 foreigners

    the space is like
    _ _ _ _
    _ _ _ _
    the foreigners sit opposite each other
    f1 _ _ _
    f2 _ _ _

    or

    f2 _ _ _
    f1 _ _ _

    there are 2 ways of arranging the foreigners ; 2!


    the foreigners can sit opposite each other in four ways

    the six natives can be arrange in 6! ways to fill the six spaces left

    thus the answer is 8!- (2! x 4 x 6!) = 34560

    b)

    I am having problems with this one

    my idea is to find out the number of ways in which the two foreigners are seated together and then subtract this from the total number of ways of them seating with no restriction.

    Suppose the two foreigners are considered as a single entity. the two foreigners can be arranged in 2! ways. Six natives can be arranged in 6! ways

    thus; 2! x 6! ways for the two foreigners can sit together

    now to find the real answer we do

    8! - (2! x 6!) = 38880

    but the book gives an answer of 31680

    Please help me what did i do wrong for part b
    Last edited by bigmansouf; Jun 15th 2019 at 10:55 AM.
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  2. #2
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    Re: Permutation Question

    a) is correct

    There are 6 slots of adjacent seats, and 2 ways of arranging the foreigners in those slots.

    So we arrive at $8! - 12\cdot 6! = 6!(56-12) = 31680$ arrangements
    Thanks from bigmansouf
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