Circle permutation necklace

In summary, the problem is asking how many necklaces can be made with 5 white beads and 5 black beads. The solution involves using circular permutation and accounting for symmetries. After some analysis, the final answer is 16.
  • #1
jameshaley
5
0

Homework Statement



How many necklace with 5 white beads and 5 black beads can be constructed?


Homework Equations



Circular Permutation problem

The Attempt at a Solution

]

I did 10!/5!5!=252

but from there I didn't get anywhere.

I know this includes repeats from rotational symmetry and reflections. but i am not really sure how to get rid of these.

i try dividing by 10 but it gives 25.2, which does not make sense to me.
 
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  • #2
There is no simple way to analyse this sort of problem. You need to break it down into cases according to the symmetries.
You know there must somewhere be a black and a white adjacent, so you could fix on such a pair. That gets you down to 8-choose-4 immediately. Then it's a matter of removing duplicates.
 
  • #3
Thinking some more about this... consider how many symmetries any given pattern might have. If only one (i.e. the identity) how many times will the given unique pattern be counted in your 252? What if two symmetries in the group? Etc. Then it's a matter of figuring how many patterns have each of the symmetry counts.
Fwiw, I make the final answer 16.
 

What is a circle permutation necklace?

A circle permutation necklace is a type of necklace that consists of a circular chain with a set of objects or beads attached to it in a specific order. The order of the objects can be rearranged by rotating the necklace, resulting in different arrangements or permutations.

How does a circle permutation necklace work?

A circle permutation necklace works by using the principles of permutation, which is the rearrangement of objects in a specific order. The objects are attached to a circular chain, and by rotating the necklace, the objects are rearranged, creating different permutations.

What are the applications of a circle permutation necklace?

Circle permutation necklaces have various applications in mathematics and computer science, such as in coding theory, cryptography, and group theory. They are also used in games and puzzles, as well as in jewelry and fashion design.

Can a circle permutation necklace have repeated objects?

Yes, a circle permutation necklace can have repeated objects. The number of permutations depends on the number of objects and their arrangement. If there are repeated objects, the number of possible permutations will be fewer than if all the objects were unique.

What are some examples of circle permutation necklaces?

Some examples of circle permutation necklaces include bead necklaces, charm bracelets, and puzzle rings. They can also be seen in various cultural and religious traditions, such as prayer beads and rosaries, which have specific arrangements of beads that can be rotated and rearranged.

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