AMC12 2019 B

AMC12 2019 B · Q11

AMC12 2019 B · Q11. It mainly tests Basic counting (rules of product/sum), Geometry misc.

How many unordered pairs of edges of a given cube determine a plane?

一个给定的立方体有多少对无序边确定一个平面？

(A) 12 12

(B) 28 28

(D) 42 42

(E) 66 66

Answer

Correct choice: (D)

正确答案：（D）

Solution

Each of the 12 edges of the cube shares a face with each of 6 other edges and is parallel to 1 additional edge. The other 4 edges are skew to the given edge. Thus each of the 12 edges can be paired with any of 7 others to determine a plane. After taking into account the double-counting of each pair, it follows that the number of unordered pairs of edges that determine a plane is $\frac{1}{2} \cdot 12 \cdot 7 = 42$.

立方体的12条边中，每条边与6条共享面的边和1条平行的额外边共面。其他4条边与给定边斜交。因此，每条12条边中的边可以与7条其他边配对确定一个平面。考虑每对边的双重计数后，确定平面的无序边对数为 $\frac{1}{2} \cdot 12 \cdot 7 = 42$。

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