Question 1132666: In how many ways can 7 charms be placed in a bracelet which has no clasp?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe it's going to be 7 factorial ways.
that would be 7 * 6 * 5 * 4 * 3 * 2 * 1 ways.
consider 3 beads.
that would be 3 factorial ways = 3 * 2 * 1 = 6 ways.
label the beads a,b,c.
the 6 ways would be abc, acb, bac, bca, cab, cba.
4 beads would be 4 * 3 factorial = 4 * 6t = 24 ways.
5 beads would be 5 * 4 factorial = 5 * 24 = 120 ways.
6 beads would be 6 * 5 factorial = 6 * 120 = 720 ways.
7 beads would be 7 * 6 factorial = 7 * 720 = 5040 ways.
the factorial symbol is !.
7 factorial would be shown as 7!.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
From the dictionary
https://dictionary.cambridge.org/us/dictionary/english/bracelet
https://dictionary.cambridge.org/us/dictionary/english/bracelet
you can read that a bracelet is a piece of jewelry that is worn around the wrist or arm.
In this definition, it is important for us now that a bracelet has a circular form like a closed line.
So, the question can be EQUIVALENTLY reformulated in THIS WAY
In how many ways can 7 charms be placed along a circumference of a circle ?
In such problems the placements that obtained one from the other by rotation of a circle by some angle are considered as INDISTINGUISHABLE.
Therefore, with each concrete placement, 6 others that obtained from the original placement by rotation, are considered as EQUIVALENT.
Therefore, the number of all possible DISTINGUISHABLE placements of 7 charms in a bracelet is  = 6! = 1*2*3*4*5*6 = 720. ANSWER
It is not 7!, as it would be in linear case !
Solved.
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The lesson to learn from my post is THIS :
This problem is the same as other classic formulation:
In how many ways 7 people can be sitting around a circular table ?
See the lesson
- Persons sitting around a cicular table
in this site.
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