AMC10 2015 B

AMC10 2015 B · Q16

AMC10 2015 B · Q16. It mainly tests Probability (basic), Divisibility & factors.

Al, Bill, and Cal will each randomly be assigned a whole number from 1 to 10, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?

Al、Bill 和 Cal 每个人将被随机分配一个从 1 到 10（包含）的整数，且三人得到的数互不相同。Al 的数是 Bill 的数的整数倍且 Bill 的数是 Cal 的数的整数倍的概率是多少？

(A) $\frac{9}{1000}$ $\frac{9}{1000}$

(B) $\frac{1}{90}$ $\frac{1}{90}$

(D) $\frac{1}{72}$ $\frac{1}{72}$

(E) $\frac{2}{121}$ $\frac{2}{121}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

There are 9 assignments satisfying the condition: (4,2,1), (6,2,1), (8,2,1), (10,2,1), (6,3,1), (9,3,1), (8,4,1), (10,5,1), and (8,4,2). There are $10 \cdot 9 \cdot 8 = 720$ possible assignments, so the probability is $\frac{9}{720} = \frac{1}{80}$.

满足条件的分配有 9 种：(4,2,1)、(6,2,1)、(8,2,1)、(10,2,1)、(6,3,1)、(9,3,1)、(8,4,1)、(10,5,1) 和 (8,4,2)。可能的分配总数为 $10 \cdot 9 \cdot 8 = 720$，因此概率为 $\frac{9}{720} = \frac{1}{80}$。

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