AMC10 2004 B

AMC10 2004 B · Q23

AMC10 2004 B · Q23. It mainly tests Basic counting (rules of product/sum), Probability (basic).

Each face of a cube is painted either red or blue, each with probability $\frac{1}{2}$. 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?

一个立方体的每个面被涂成红色或蓝色，每种颜色概率均为 $\frac{1}{2}$，且每个面的颜色独立确定。涂色后的立方体可以放置在水平面上，使得四个垂直面全为同一颜色的概率是多少？

(A) \frac{1}{4} \frac{1}{4}

(B) \frac{5}{16} \frac{5}{16}

(D) \frac{7}{16} \frac{7}{16}

(E) \frac{1}{2} \frac{1}{2}

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) If the orientation of the cube is fixed, there are $2^6 = 64$ possible arrangements of colors on the faces. There are $2\binom{6}{6} = 2$ arrangements in which all six faces are the same color and $2\binom{6}{5} = 12$ arrangements in which exactly five faces have the same color. In each of these cases the cube can be placed so that the four vertical faces have the same color. The only other suitable arrangements have four faces of one color, with the other color on a pair of opposing faces. Since there are three pairs of opposing faces, there are $2(3) = 6$ such arrangements. The total number of suitable arrangements is therefore $2 + 12 + 6 = 20$, and the probability is $20/64 = 5/16$.

(B) 如果立方体的朝向固定，则面上颜色的排列共有 $2^6 = 64$ 种可能。共有 $2\binom{6}{6} = 2$ 种排列使得六个面颜色都相同，以及 $2\binom{6}{5} = 12$ 种排列使得恰好五个面颜色相同。在这两种情况下，都可以摆放立方体使四个侧面（竖直面）颜色相同。唯一其他合适的排列是：四个面为一种颜色，另一种颜色出现在一对相对的面上。由于相对面的配对有三对，因此这样的排列有 $2(3) = 6$ 种。故合适排列总数为 $2 + 12 + 6 = 20$，概率为 $20/64 = 5/16$。

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