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What is the number of different pairs of parallel edges on a cubical

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What is the number of different pairs of parallel edges on a cubical [#permalink]

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New post 04 Dec 2017, 03:38
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What is the number of different pairs of parallel edges on a cubical [#permalink]

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New post 04 Dec 2017, 09:56
Bunuel wrote:
What is the number of different pairs of parallel edges on a cubical wooden block?

(A) 18
(B) 12
(C) 8
(D) 6
(E) 4

Attachment:
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zzzzzzzz.png [ 8.67 KiB | Viewed 361 times ]

There are a few ways to approach this problem.
I drew the first one. It seems the easiest to me.

Sets of parallel edges * 3
See diagram:
There are three sets of parallel edges (purple, green, pink)

Each edge is parallel to three other edges in three directions: top, bottom, diagonally across*

Without duplicates:
Purple edge A is parallel to B, to D, and to C
Purple edge B is parallel to C and D
Purple edge C is parallel to D

Pairs of parallel lines for purple edges?

(A-B), (A-C), (A-D), (B-C), (B-D), and (C-D) = 6 pairs of parallel lines

Pairs of parallel lines for pink edges and green edges will be the same:
So 3 sets * 6 pairs = 18 pairs

Answer A

Number of edges times number of pairs divided by 2

Each edge is parallel to three other edges (see above)
There are 12 edges with 3 pairs of parallel lines
12 * 3 = 36 pairs, but
That figure has double counting: 36/2 = 18 pairs of parallel lines

Answer A

Combinations
Someone else can explain; I can see it but I can't explain it well. :|
There are three edges: vertical, horizontal, and, in the diagram, "connecting" or "projecting" (backwards into 3-D perspective):
\(3 * (_4C_ 2) = 3 * 6 = 18\)

Answer A

*Regarding edges that are parallel on the diagonal plane: the top left edge is parallel to the bottom right edge, e.g.: you could insert a plane on the diagonal / slant. The lines will never meet and the distance between any two corresponding points on the two lines will always be the same. They are parallel edges on the same plane (the diagonal plane of symmetry).
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Re: What is the number of different pairs of parallel edges on a cubical [#permalink]

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New post 04 Dec 2017, 10:48
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A cube has 12 edges. There are 3 *Groups* of parallel edges

Each Group therefore has 4 edges

Lets take Group 1:
We need pairs of parallel edges. Which means we need to pick 2 edges from 4 edges.
This is simply 4C2 = \(\frac{4 x 3}{2 x 1}\) = 6

Therefore Group 2 and Group 3 will also have the same = 6 and 6

6 + 6 + 6 = 18 total "pairs" of parallel edges.
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Re: What is the number of different pairs of parallel edges on a cubical   [#permalink] 04 Dec 2017, 10:48
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