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Math Help - Help to solve a differnsial text problem

  1. #1
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    Help to solve a differnsial text problem

    You have invited some friends for dinner and decide to give different color napkins to the people sitting next to each other. You wonder how many ways this can be done when have napkins in four different colors.

    Let a_n be the number of ways to place the napkins on if there are n people (with fixed seats) around the table. Show first that a_2 = 12 and a_3 = 24 How many such locations of napkins available if there 11 round table?

    Hint: Make a difference equations for a_n and solve this.
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  2. #2
    MHF Contributor

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    Re: Help to solve a differnsial text problem

    Let a(n) be the number of ways to lay out napkins for n people. Now, suppose you add one more person. You have four colors of napkin to give that person but you cannot give him the napkins of the people on either side. Since those are of different colors (the people were sitting next to each other) you can choose either of the two remaining colors to give him. That is, a(n+ 1)= a(n)+ 2.
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