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Math Help - circular permutation

  1. #1
    Senior Member Sampras's Avatar
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    circular permutation

    Suppose there are  n families with  5 people in each family for a total of  5n people arranged around a circular table. How many ways are there to seat the people such that each person sits next to another member of his family?

    So fix  2 people from each family (treat them as a unit). This leaves  (5n-2n)! = 3n! ways to arrange the rest?
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  2. #2
    Senior Member Shanks's Avatar
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    You answer is definitely false.
    just take n=1 as a counterexample for your solution.
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  3. #3
    Senior Member Shanks's Avatar
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    Hint:for a family, there is only two way to be seated:
    (1)all the members are seated togather as a unit.
    (2)all the members are divided into two group consisting of 2 and 3 members, members in each group are seated togather as a unit, and two units are seperated by members of other family.
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  4. #4
    Senior Member Sampras's Avatar
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    Quote Originally Posted by Shanks View Post
    Hint:for a family, there is only two way to be seated:
    (1)all the members are seated togather as a unit.
    (2)all the members are divided into two group consisting of 2 and 3 members, members in each group are seated togather as a unit, and two units are seperated by members of other family.
    So seat one family and there are  (n-1)! ways of arranging the other members.
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  5. #5
    Senior Member Shanks's Avatar
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    No, It is possible that some fimalies are seperated into two Units(groups) by other family.
    for example, here is a example of n=2 to clearify my posted thread:
    (1)No family is seperated: that is all members in the same family are seated togather as a unit.
    (2)For the case n=2, only one family are seperated is equivalent to (1). If
    n>2, it is possible that one or some families are seperated by other family. We need a arguement on the number of families that are seperated.
    (3)Two families are seperated, each family are divided into two groups consisting of 2 and 3 members.
    Denote the two family members by A_1,...,A_5; B_1,...,B_5.
    Here is a example that all members are seated in a circle clockwisely:
    A_1,A_2,B_1,B_2,A_3,A_4,A_5,B_3,B_4,B_5
    similarly, You can find all the ways that all members are seated in a circle such that each member sits next to a member of his family.
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