Difference between revisions of "2019 AMC 12B Problems/Problem 11"
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<math>\textbf{(A) } 12 \qquad \textbf{(B) } 28 \qquad \textbf{(C) } 36\qquad \textbf{(D) } 42 \qquad \textbf{(E) } 66</math> | <math>\textbf{(A) } 12 \qquad \textbf{(B) } 28 \qquad \textbf{(C) } 36\qquad \textbf{(D) } 42 \qquad \textbf{(E) } 66</math> | ||
| − | ==Solution== | + | ==Solution 1== |
WLOG, pick one of the 12 edges of the cube to be among the two selected. We seek the answer by computing the probability that a random choice of second edge satisfies the problem statement. | WLOG, pick one of the 12 edges of the cube to be among the two selected. We seek the answer by computing the probability that a random choice of second edge satisfies the problem statement. | ||
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//Can somebody better with latex than me put a picture in here? | //Can somebody better with latex than me put a picture in here? | ||
| + | |||
| + | ==Solution 2== | ||
| + | Case 1: The two edges are on the same face. There are <math>6 \cdot {4 \choose 2}=36</math> possibilities. | ||
| + | |||
| + | Case 2: The two edges are parallel but not on the same face. There are <math>6</math> possibilities. | ||
| + | |||
| + | <math>36 + 6 = \boxed{\textbf{(D) }42}</math> | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2019|ab=B|num-b=10|num-a=12}} | {{AMC12 box|year=2019|ab=B|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 21:27, 14 February 2019
Contents
Problem
How many unordered pairs of edges of a given cube determine a plane?
Solution 1
WLOG, pick one of the 12 edges of the cube to be among the two selected. We seek the answer by computing the probability that a random choice of second edge satisfies the problem statement.
For two line segments in space to correspond to a common plane, they must correspond to lines that either intersect or are parallel. If all 12 line segments are extended to lines, our first edge's line intersects 4 lines and is parallel to another 3. Thus 7 of the 11 line segments satisfy the problem statement.
We compute: 
//Can somebody better with latex than me put a picture in here?
Solution 2
Case 1: The two edges are on the same face. There are 
possibilities.
Case 2: The two edges are parallel but not on the same face. There are 
possibilities.
See Also
| 2019 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 10 |
Followed by Problem 12 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
