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Discrete math-counting permutation

  1. Nov 28, 2006 #1

    cak

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    Can someone help me understand this one? The problem is: Four beads-red,blue,yellow, and green-are arranged on a string to make a simple necklace as shown in the figure. How many arrangements are possible?

    The answer in the book is 3, but I don't get it.

    I thought it woud be a permutation because order does matter. So I assume that since there are 4 objects and 4 blanks to fill the permutation counting formula would be (4!)/(4-4)!
    Therefore this would equal 4*3*2*1/1=24

    The book says the answer is three...am I doing something wrong???

    HELLLP :yuck:
     
  2. jcsd
  3. Nov 28, 2006 #2
    Remember the arrangement Red, blue, yellow, green is exactly the same as the arrangement green, red, blue, yellow. Why? Because it's a necklace. It's circular. There really is not "first" element. Since it's a circle, you can call any element the first element you want. It doesn't change the order.


    If it's still unclear, take a simpler example. Say you only have 2 items on your necklace, a car and a house (for whatever reason). How many different arrangements do you have?

    Well we have car, house
    and we have house,car
    except house, car is exactly the same as car, house. We can just shift the necklace a little bit so we start counting the house first. It doesn't actually change the order or arrangement of it.
     
    Last edited: Nov 28, 2006
  4. Nov 29, 2006 #3

    HallsofIvy

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    Staff Emeritus
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    To continue what KoGs was saying: If you had the beads in a line, then
    RBYG would be different from BYGR. But if the beads are on a circular string, making a necklace, then RBYG and BYGR would be exactly the same- you just started listing from a different point. Do this: first select any one of the beads, say R, to put on the necklace. NOW how many choices do you have for the second bead to put on? The third? The fourth? How many choices did you have altogether?
     
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