How many choices do we have to make a necklace from 8 beads of different colors?

Is this a homework question?

Choose the colors, then begin a pattern.

Sarah

Quote:

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Originally Posted by deepthi03

How many choices do we have to make a necklace from 8 beads of different colors?

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If you were stringing N beads on a straight wire left-to-right this would be quite straight forward - you would have N choices for the 1st bead, then N-1 for the 2nd, N-2 for the 3rd etc. for a total of N! Choices. But if this necklace is circular, so that you can slide the beads around the loop it's a little more complicated, because from any one bead pattern you get essentially N different "arrangements." Thus all of the following 8 arrangements are really identical, counting clockwise from the first bead position:

ABCDFEGH
BCDEFGHA
CDEFGHAB
DEFGHABC
EFGHABCD
FGHABCDE
GHABCDEF
HABCDEFG

Hence the number of unique ways to arrange the beads is N! Divided by N. But that's not all, because if you turn the necklace over and wear it backwards you see that for any bead pattern there are two arrangements possible. In other words, the following two arrangemenst are identical:

ABCDEFGH and AHGFEDCB

So you have to divide N! By N and then by 2 to get the number of truly unique arrangements on a circular necklace (if you're allowed to slide beads around the loop and turn it over).