Difference between revisions of "2009 AMC 10A Problems/Problem 24"

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== Problem ==
 
== 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?
 
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?
  
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</math>
 
</math>
  
== Solution 1 ==
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== Solutions ==
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=== Solution 1 ===
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First of all, number of planes determined by any three vertices of a cube is <math>20</math> (<math>6</math> surface, <math>6</math> opposing parallel edges, <math>8</math> points cut by three remote vertices). Among these <math>20</math> planes only <math>6</math> surfaces will not cut into the cube.
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Secondly, to choose three vertices randomly, the four vertices planes each will be chosen <math>4</math> times, while the three vertices planes each will be chosen once.
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To conclude, the probability of a cutting in plane is <math>(6\cdot 4+8\cdot 1)/(12\cdot 4+8\cdot 1) = 32/56 = 4/7</math> (C)
  
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-Vader10,Oct.6 2020-
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=== Solution 2 ===
 
We will try to use symmetry as much as possible.
 
We will try to use symmetry as much as possible.
  
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</cmath>
 
</cmath>
  
== Solution 2 ==
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=== Solution 3 (Cheap solution same approach as Solution 2) ===
  
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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 <math>7</math> remaining vertices, <math>4</math> 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 <math>4/7</math>.
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=== Solution 4 ===
 
There are <math>\binom{8}{3}=56</math> 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 <math>\binom{8}{3}=56</math> 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 four ways to choose three points from the vertices of a single face. Since there are six faces, <math>4 \times 6 = 24</math>.
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There are <math>\binom{4}{3}=4</math> ways to choose three points from the vertices of a single face. Since there are six faces, <math>4 \times 6 = 24</math>.
  
 
Thus, the probability of what we don't want is <math>\frac{24}{56} = \frac{3}{7}</math>. Using complementary probability,  
 
Thus, the probability of what we don't want is <math>\frac{24}{56} = \frac{3}{7}</math>. Using complementary probability,  
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1- \frac 37 = \boxed{\frac 47}
 
1- \frac 37 = \boxed{\frac 47}
 
</cmath>
 
</cmath>
 
 
  
 
== See Also ==
 
== See Also ==

Revision as of 00:38, 19 October 2020

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 $(6\cdot 4+8\cdot 1)/(12\cdot 4+8\cdot 1) = 32/56 = 4/7$ (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,

\[1- \frac 37 = \boxed{\frac 47}\]

See Also

2009 AMC 10A (ProblemsAnswer KeyResources)
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
All AMC 10 Problems and Solutions

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