AMC 8 2008 Problem 19

AMC8 2008 年第 19 题

概率★★★☆☆
Eight points are spaced around at intervals of one unit around a 2×22 \times 2 square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?
如图所示，八个点绕着一个 2×2 的正方形等间距分布，相邻点的距离均为 1 个单位。从这 8 个点中随机选择其中两个点，则这两点相距一个单位的概率是多少？
AMC 8 2008 Problem 19 figure
  1. A.14\displaystyle \frac{1}{4}
  2. B.27\displaystyle \frac{2}{7}
  3. C.411\displaystyle \frac{4}{11}
  4. D.12\displaystyle \frac{1}{2}
  5. E.47\displaystyle \frac{4}{7}

正确答案 · Correct Answer

B

解析 · Solution

Solution 1

The two points are one unit apart at 8 places around the edge of the square. There are (82)=28\binom{8}{2} = 28 ways to choose two points. The probability is 828=27\displaystyle \frac{8}{28} = \frac27

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