I'm trying to calculate the number of unique bracelets of length $6$ that can be made from $11$ differently colored beads.
However there is one important key detail! Each bracelet has to contain exactly one bead of an unique color, $X$ (e.g. gold), that is not used in the other beads. The remaining $5$ other beads can have any of the $10$ differently colored beads ($0$-$9$ (e.g. blue, brown, grey, white, etc...) can be repeated and used as many times as possible as the length permits.
Therefore the beads generated would be include for example:
My initial attempt:
I reasoned that this bracelet problem can actually be solved through calculating necklaces.
Calculating the number of necklaces for $n=5$, $k=10$ yields $20008$ unique necklaces starting from:
Then to generate unique bracelets from these necklaces, I have three options to place the $X$ bead for each .
Original Necklace = $00001$
- Unique Bracelets Generated = $X00001$, $0X0001$, $00X001$
Then I would remove any non-unique bracelets. From my thoughts, the only duplicate bracelets would be those generated from necklaces with only a single color. For this example, I would remove $10$ (number of colors) * $2$ (duplicate sets) = $20$ bracelets.
- Original Necklace = $000000$
- Unique Bracelets Generated = $X00000$
- (not $0X0000$ or $00X000$ since they are the same as $X00000$)
Therefore from this reasoning, I'm guessing my solution would be simply $20008*3 - 20$ unique bracelets = $60014$ bracelets.
My questions are:
Is this a feasible or correct way of thinking about the problem?
Is this the correct answer?
Is there a better method to solve this problem?