2019 AMC 12B Problem 11

Below is the professionally curated solution for Problem 11 of the 2019 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12B solutions, or check the answer key.

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Concepts:cube geometry 3D geometry counting pairs

Difficulty rating: 1640

11.

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

$12$

$28$

$36$

$42$

$66$

Solution:

Two edges determine a plane exactly when they are coplanar, that is, parallel or intersecting.

The $12$ edges split into $3$ directions of $4$ parallel edges, giving $3\binom{4}{2}=18$ parallel pairs. Edges sharing a vertex give $8\binom{3}{2}=24$ intersecting pairs.

The total is $18+24=42.$

Thus, D is the correct answer.

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