AMC12 2000 A

AMC12 2000 A · Q25

AMC12 2000 A · Q25. It mainly tests Permutations, Counting with symmetry / Burnside (rare).

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

用八个全等的正三角形（每个颜色都不同）构造一个正八面体。有多少种可区分的方式来构造该八面体？（若两个彩色八面体无法通过旋转使其看起来完全相同，则它们是可区分的。）

(A) 210 210

(B) 560 560

(D) 1260 1260

(E) 1680 1680

Answer

Correct choice: (E)

正确答案：（E）

Solution

Since the octahedron is indistinguishable by rotations, without loss of generality fix a face to be red. There are $7!$ ways to arrange the remaining seven colors, but there still are three possible rotations about the fixed face, so the answer is $7!/3 = 1680$.

由于八面体在旋转下不可区分，不失一般性，固定一个面为红色。其余七种颜色有 $7!$ 种排列方式，但绕固定的这个面仍有三种可能的旋转，因此答案为 $7!/3 = 1680$。

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