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Math Help - Ways of seating people around a circular table

  1. #1
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    Exclamation Ways of seating people around a circular table

    Six people sit around a circular table. Person A and B must not sit next to each other...how many ways are there of seating them?
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  2. #2
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    Hello, HelenaStage!

    Six people sit around a circular table.
    Person A and B must not sit next to each other.
    How many ways are there of seating them?

    With 6 people around a circular table,
    . . there are: . 5! \:=\:120 ways to seat them.

    Duct-tape A and B together
    . . Then we have 5 "people" to arrange: . \boxed{AB}\;C\;D\;E\:F
    There are: . 4! = 24 ways to seat them.

    But the 5 "people" could be: . \boxed{BA}\;C\;D\;E\;F

    Hence, there are: . 2 \times 24 \:=\:48 ways that A and B are adjacent.


    Therefore, there are: . 120 - 48 \:=\:72 ways in which A and B are not adjacent.

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  3. #3
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    Quote Originally Posted by HelenaStage View Post
    Six people sit around a circular table. Person A and B must not sit next to each other...how many ways are there of seating them?
    Here is a different and direct approach.
    Seat A at the table. There are then three places to seat B.
    The other four can be seated in 4! ways.
    3(4!)=72
    Last edited by Plato; November 14th 2009 at 11:52 AM.
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  4. #4
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    Hello, Plato!

    Elegant solution! . . . I like it!

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