How many different necklaces can be formed with $6$ white and $5$ red beads?

Since total number of beads is $11$ according to me it should be $\dfrac{11!}{6!5!}$ but correct answer is $21$. How does that come?

  • $\begingroup$ since the necklace is "circular" there are many repeated similar combinations in $\dfrac{11!}{6!5!}$ $\endgroup$ – Dylan Zammit Mar 14 '18 at 7:19
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    $\begingroup$ By my reckoning, the answer is $26$. $\endgroup$ – Lord Shark the Unknown Mar 14 '18 at 8:54
  • $\begingroup$ @LordSharktheUnknown I get $26$ for a bracelet, which is invariant under both rotation and reflection. However, by convention, necklaces are only invariant under rotation. $\endgroup$ – N. F. Taussig Mar 14 '18 at 13:02
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    $\begingroup$ @N.F.Taussig Then in that case, the answer is 42. $\endgroup$ – Lord Shark the Unknown Mar 14 '18 at 13:12

First consider $$\dfrac{11!}{5!6!}$$ as you have done. Now since the necklace is "circular" you must divide by the number of beads to remove the repeated arrangements, that is:$$\dfrac{11!}{5!6!11}$$

Also, the necklace can be flipped over, thus twice as many combinations had been counted so far, so you must divide by 2. That is:$$\dfrac{11!}{5!\cdot6!\cdot11 \cdot 2}=21$$

  • $\begingroup$ But some necklaces are preserved by the "flipping over" operation. $\endgroup$ – Lord Shark the Unknown Mar 14 '18 at 7:24
  • $\begingroup$ Are you referring to the cases where the beads are "symmetrical"...such as: rrwwwrr ? $\endgroup$ – Dylan Zammit Mar 14 '18 at 7:28
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    $\begingroup$ Exactly, one needs some (slightly) non-trivial application of Burnside's lemma. $\endgroup$ – Lord Shark the Unknown Mar 14 '18 at 7:35
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    $\begingroup$ According to en.wikipedia.org/wiki/Necklace_(combinatorics), the convention in combinatorics is that a "necklace" cannot be flipped over. If it can be flipped, it is called a "bracelet". $\endgroup$ – awkward Mar 14 '18 at 13:03

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