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Math Help - Finding the number of ways

  1. #1
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    Finding the number of ways

    Hi, if we have an initial arrangement say 12345 where all elements are unique, then in how many ways can one re-arrange them so that in the final configurations none of the elements which were adjacent to each other in the given arrangement are adjacent now in the final configurations.?
    Does this involve some series?
    E.g for 5 element say 12345, the number of such arrangements is 14.
    Thanks.
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  2. #2
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    Re: Finding the number of ways

    Quote Originally Posted by pranay View Post
    Hi, if we have an initial arrangement say 12345 where all elements are unique, then in how many ways can one re-arrange them so that in the final configurations none of the elements which were adjacent to each other in the given arrangement are adjacent now in the final configurations.? Does this involve some series?
    E.g for 5 element say 12345, the number of such arrangements is 14.
    How did you get that 14?

    Note that for 123 there are no cases.

    How many are there for 1234?
    Is this one of those, 3142~?
    Given that string, there are three places to put a 5: 53142,~35142,~31425.

    So if we know how may valid cases there are for 1234 we can build valid cases for 12345.

    I will let you work on that.
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  3. #3
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    Re: Finding the number of ways

    if the initial configuration is ABCDE , then the 14 required configurations are:
    ACEBD
    ADBEC
    BDACE
    BDAEC
    BECAD
    CADBE
    CAEBD
    CEADB
    CEBDA
    DACEB
    DBEAC
    DBECA
    EBDAC
    ECADB
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