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Math Help - easy, easy question

  1. #1
    Member billym's Avatar
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    easy, easy question

    Can somebody please help me out with this? I had the answer easily the first time, but now my mind is blank:

    How many ways are there to seat 10 people, consisting of 5 couples, in a row of seats (10 seats wide) if all couples are to get adjacent seats?
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  2. #2
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    Hello, billym!

    How many ways are there to seat 10 people, consisting of 5 couples,
    in a row of seats (10 seats wide) if all couples are to get adjacent seats?
    For reference, let the couples be: . (A,a),\:(B,b),\:(C,c),\:(D,d),\:(E,e)


    Duct-tape the couples together.

    We have 5 "people" to arrange: . \boxed{Aa}\;\boxed{Bb}\;\boxed{Cc}\;\boxed{Dd}\;\b  oxed{Ee}

    There are: . 5! \,=\,120 permutations.


    But for each permutation, the couples can be "swtiched".

    . . \boxed{Aa} could be \boxed{aA}, \boxed{Bb} could be \boxed{bB}, and so on.

    There are: . 2^5 \,=\,32 possible switchings.


    Therefore, there are: . 120\cdot32 \:=\:{\color{blue}3840} seating arrangements.

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, billym!

    For reference, let the couples be: . B,b),\C,c),\D,d),\E,e)" alt="(A,a),\B,b),\C,c),\D,d),\E,e)" />


    Duct-tape the couples together.

    I've always suspected you had an evil streak, Soroban!

    I have a friend who swears you can do anything with duct-tape. Now I know you can even solve math problems with it!

    We have 5 "people" to arrange: . \boxed{Aa}\;\boxed{Bb}\;\boxed{Cc}\;\boxed{Dd}\;\b  oxed{Ee}

    There are: . 5! \,=\,120 permutations.


    But for each permutation, the couples can be "swtiched".

    . . \boxed{Aa} could be \boxed{aA}, \boxed{Bb} could be \boxed{bB}, and so on.

    There are: . 2^5 \,=\,32 possible switchings.


    Therefore, there are: . 120\cdot32 \:=\:{\color{blue}3840} seating arrangements.
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  4. #4
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    Hello, HallsofIvy!

    I have a friend who swears you can do anything with duct tape.
    Duct tape is like The Force.

    It has a light side and dark side
    and it holds the universe together.

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