# 2015 AMC 8 Problems/Problem 12

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?

$\textbf{(A) }6 \quad\textbf{(B) }12 \quad\textbf{(C) } 18 \quad\textbf{(D) } 24 \quad \textbf{(E) } 36$ $[asy] import three; currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); label("D",(0,0,0),S); label("A",(0,0,1),N); label("H",(0,1,0),S); label("E",(0,1,1),N); label("C",(1,0,0),S); label("B",(1,0,1),N); label("G",(1,1,0),S); label("F",(1,1,1),N); [/asy]$

## Solution 1

We first count the number of pairs of parallel lines that are in the same direction as $\overline{AB}$. The pairs of parallel lines are $\overline{AB}\text{ and }\overline{EF}$, $\overline{CD}\text{ and }\overline{GH}$, $\overline{AB}\text{ and }\overline{CD}$, $\overline{EF}\text{ and }\overline{GH}$, $\overline{AB}\text{ and }\overline{GH}$, and $\overline{CD}\text{ and }\overline{EF}$. These are $6$ pairs total. We can do the same for the lines in the same direction as $\overline{AE}$ and $\overline{AD}$. This means there are $6\cdot 3=\boxed{\textbf{(C) } 18}$ total pairs of parallel lines.

## Solution 2

Pick a random edge. Given another edge, the probability that it is parallel to this edge is $\frac{3}{12-1}=\frac{3}{11}$. Keep in mind we already used one edge. There are $12$ edges so $\binom{12}{2}=66$ pairs. So our answer is $\frac{3}{11} \times 66=\boxed{\textbf{(C)}~18}$.