2009 AMC 10A Problems/Problem 24

Problem

Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?

$\mathrm{(A)}\ \frac{1}{4} \qquad \mathrm{(B)}\ \frac{3}{8} \qquad \mathrm{(C)}\ \frac{4}{7} \qquad \mathrm{(D)}\ \frac{5}{7} \qquad \mathrm{(E)}\ \frac{3}{4}$

Solutions

Solution 1

First of all, number of planes determined by any three vertices of a cube is $20$ ( $6$ surface, $6$ opposing parallel edges, $8$ points cut by three remote vertices). Among these $20$ planes only $6$ surfaces will not cut into the cube. Secondly, to choose three vertices randomly, the four vertices planes each will be chosen $4$ times, while the three vertices planes each will be chosen once. To conclude, the probability of a cutting in plane is $\frac {(6\cdot 4+8\cdot 1)}{(12\cdot 4+8\cdot 1)} = \frac {32}{56}$ = $\boxed {\textbf{(C)}}$ .

-Vader10,Oct.6 2020

Solution 2

We will try to use symmetry as much as possible.

Pick the first vertex $A$ , its choice clearly does not influence anything.

Pick the second vertex $B$ . With probability $3/7$ vertices $A$ and $B$ have a common edge, with probability $3/7$ they are in opposite corners of the same face, and with probability $1/7$ they are in opposite corners of the cube. We will handle each of the cases separately.

In the first case, there are $2$ faces that contain the edge $AB$ . In each of these faces there are $2$ other vertices. If one of these $4$ vertices is the third vertex $C$ , the entire triangle $ABC$ will be on a face. On the other hand, if $C$ is one of the two remaining vertices, the triangle will contain points inside the cube. Hence in this case the probability of choosing a good $C$ is $2/6 = 1/3$ .

In the second case, the triangle $ABC$ will not intersect the cube if point $C$ is one of the two points on the side that contains $AB$ . Hence the probability of $ABC$ intersecting the inside of the cube is $2/3$ .

In the third case, already the diagonal $AB$ contains points inside the cube, hence this case will be good regardless of the choice of $C$ .

Summing up all cases, the resulting probability is: $\frac 37\cdot\frac 13 + \frac 37\cdot \frac 23 + \frac 17\cdot 1 = \boxed{\frac 47}$

Solution 3 (Cheap solution, same approach as Solution 2)

This problem can be approached the same way, by picking vertices, but with a much faster and kind of cheap solution: Pick any vertex A and a face it touches. For vertex B, out of the $7$ remaining vertices, $4$ of them aren't on the same face as the one chosen for vertex A, so vertex C can be placed anywhere and the plane will no matter what be in the cube. Therefore, the probability of choosing a valid vertex B is $4/7$ .

Solution 4

There are $\binom{8}{3}=56$ ways to pick three vertices from eight total vertices; this is our denominator. In order to have three points inside the cube, they cannot be on the surface. Thus, we can use complementary probability.

There are $\binom{4}{3}=4$ ways to choose three points from the vertices of a single face. Since there are six faces, $4 \times 6 = 24$ .

Thus, the probability of what we don't want is $\frac{24}{56} = \frac{3}{7}$ . Using complementary probability,

$1- \frac 37 = \boxed{\frac 47}$

Solution 5 (Casework)

This problem is fairly simple. Start with $2$ points WLOG.

Case $1$ : You pick 2 points that are diagonally across from each other but still on the same face. This happens with probability $\frac {3}{7}$ . We notice that $4$ out of the $6$ possible outcomes include a point inside the cube. Therefore, the probability of this happening is $\frac {2}{7}$ .

Case $2$ : You pick 2 points that a directly across from each other. This happens with probability $\frac {3}{7}$ . We notice that $2$ out of the $6$ possible outcomes include a point inside the cube. Therefore, the probability of this happening is $\frac {1}{7}$ .

Case $3$ : You pick 2 points that is a space diagonal of the cube. This happens with probability $\frac {1}{7}$ . Clearly, all of the points contain a point inside the cube, so our probability is $\frac {1}{7}$ .

Adding these probabilities gives us $\boxed {\frac {4}{7}}$

~Arcticturn

(Video solution)

Video: https://youtu.be/5PiNMIxItVQ ~DaBobWhoLikeMath1234

2009 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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
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