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Difference between revisions of "2004 AMC 10B Problems/Problem 23"

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<math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{5}{16} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{7}{16} \qquad \mathrm{(E) \ } \frac{1}{2} </math>
 
<math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{5}{16} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{7}{16} \qquad \mathrm{(E) \ } \frac{1}{2} </math>
  
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==Solution==
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==Solution 1==
  
 
Label the six sides of the cube by numbers <math>1</math> to <math>6</math> as on a classic dice. Then the "four vertical faces" can be: <math>\{1,2,5,6\}</math>, <math>\{1,3,4,6\}</math>, or <math>\{2,3,4,5\}</math>.
 
Label the six sides of the cube by numbers <math>1</math> to <math>6</math> as on a classic dice. Then the "four vertical faces" can be: <math>\{1,2,5,6\}</math>, <math>\{1,3,4,6\}</math>, or <math>\{2,3,4,5\}</math>.
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Therefore
 
Therefore
 
<math>|A\cup B\cup C| = 8+8+8-2-2-2+2 = 20</math>, and the result is <math>\frac{20}{64}=\boxed{\frac{5}{16}}</math>.
 
<math>|A\cup B\cup C| = 8+8+8-2-2-2+2 = 20</math>, and the result is <math>\frac{20}{64}=\boxed{\frac{5}{16}}</math>.
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==Solution 3==
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==Solution 2 ==
  
 
Suppose we break the situation into cases that contain four vertical faces of the same color:
 
Suppose we break the situation into cases that contain four vertical faces of the same color:
  
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I. Two opposite sides of same color
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I. Two opposite sides of same color:
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There are 3 ways to choose the two sides, and then two colors possible, so <math>3*2=6</math>
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There are 3 ways to choose the two sides, and then two colors possible, so <math>3 \cdot 2=6</math>.
  
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II. One face different from all the others
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II. One face different from all the others:
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There are 6 ways to choose this face, and 2 colors, so <math>6*2=12</math>
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There are 6 ways to choose this face, and 2 colors, so <math>6 \cdot 2=12</math>.
  
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III. All faces same
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III. All faces are the same:
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There are 2 colors, so two ways for all faces to be the same.  
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There are 2 colors, and so two ways for all faces to be the same.  
  
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Adding them up, we have a total of 20 ways to have four vertical faces the same color. The are <math>2^6</math> ways to color the cube, so the answer is <math>\frac{20}{64}=\boxed{\frac{5}{16}}</math>
+
Adding them up, we have a total of <math>20</math> ways to have four vertical faces the same color. There are <math>2^6</math> ways to color the cube, so the answer is <math>\frac{20}{64}=\boxed{\frac{5}{16}}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 18:14, 11 June 2024

Problem

Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

$\mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{5}{16} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{7}{16} \qquad \mathrm{(E) \ } \frac{1}{2}$

Solution 1

Label the six sides of the cube by numbers $1$ to $6$ as on a classic dice. Then the "four vertical faces" can be: $\{1,2,5,6\}$, $\{1,3,4,6\}$, or $\{2,3,4,5\}$.

Let $A$ be the set of colorings where $1,2,5,6$ are all of the same color, similarly let $B$ and $C$ be the sets of good colorings for the other two sets of faces.

There are $2^6=64$ possible colorings, and there are $|A\cup B\cup C|$ good colorings. Thus the result is $\frac{|A\cup B\cup C|}{64}$. We need to compute $|A\cup B\cup C|$.

Using the Principle of Inclusion-Exclusion we can write \[|A\cup B\cup C| = |A|+|B|+|C| - |A\cap B| - |A\cap C| - |B\cap C| + |A\cap B\cap C|\]

Clearly $|A|=|B|=|C|=2^3=8$, as we have two possibilities for the common color of the four vertical faces, and two possibilities for each of the horizontal faces.

What is $A\cap B$? The faces $1,2,5,6$ must have the same color, and at the same time faces $1,3,4,6$ must have the same color. It turns out that $A\cap B=A\cap C=B\cap C= A\cap B\cap C =$ the set containing just the two cubes where all six faces have the same color.

Therefore $|A\cup B\cup C| = 8+8+8-2-2-2+2 = 20$, and the result is $\frac{20}{64}=\boxed{\frac{5}{16}}$.

Solution 2

Suppose we break the situation into cases that contain four vertical faces of the same color:

I. Two opposite sides of same color: There are 3 ways to choose the two sides, and then two colors possible, so $3 \cdot 2=6$.

II. One face different from all the others: There are 6 ways to choose this face, and 2 colors, so $6 \cdot 2=12$.

III. All faces are the same: There are 2 colors, and so two ways for all faces to be the same.

Adding them up, we have a total of $20$ ways to have four vertical faces the same color. There are $2^6$ ways to color the cube, so the answer is $\frac{20}{64}=\boxed{\frac{5}{16}}$.

See also

2004 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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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