Discussion
The number of distinct bracelets of five beads made up of red, blue, and green beads (two bracelets are indistinguishable if the rotation of one yield another) is,
Answer: C
Counting bracelets of five beads in three colours, where two are the same if one is a rotation of the other, is counting necklaces under the cyclic group of order 5, by Burnside's lemma. The identity fixes all \(3^5 = 243\) colourings, and each of the four non-trivial rotations fixes only the 3 single colour ones, because 5 is prime. So
\[ \frac{3^5 + 4 \times 3}{5} \]\[ = \frac{243 + 12}{5} = 51 \]The keyed answer is 51. The number 243 counts all colourings, 81 is \(3^4\) and 47 does not come from the formula.
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