PUMaC 2009 · 组合（A 组） · 第 8 题

PUMaC 2009 — Combinatorics (Division A) — Problem 8

Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy? 1

解析

Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy? Solution. 198. This uses the Polya-Burnside lemma: if the symmetries of a set object X are in a set G , then the number of objects, where two objects are the same if there is a symmetry in G that sends one to the other, is ∑ 1 g | X | | G | g ∈ G g where | X | is the number of objects fixed by g . In this case, there is the identity rotation, 7 which fixes everything(i.e. 3 bracelets), 6 rotations, each of which fix 3 bracelets apiece, and 7 4 3 +6 ∗ 3+7 ∗ 3 4 7 reflections, each of which fix 3 bracelets apiece. The total is thus = 198 total 14 bracelets. 3