Question

编号为 $1$ 到 $10$ 的十块牌面朝下放着。随机翻开一块牌，并掷一个骰子。牌上的数与骰子点数的乘积是平方数的概率是多少？

Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?

$\ \dfrac{1}{10}$

$\ \dfrac{1}{6}$

$\ \dfrac{11}{60}$

$\ \dfrac{1}{5}$

$\ \dfrac{7}{30}$

答案：C

知识点：基本概率完全平方数分类讨论

难度评级：1630

解答：

如果骰子点数为 $1$ ，牌可以是 $1,4,9$ ，有 $3$ 种组合。

如果骰子点数为 $2$ ，牌可以是 $2,8$ ，有 $2$ 种组合。

如果骰子点数为 $3$ ，牌可以是 $3$ ，有 $1$ 种组合。

如果骰子点数为 $4$ ，牌可以是 $1,4,9$ ，有 $3$ 种组合。

如果骰子点数为 $5$ ，牌可以是 $5$ ，有 $1$ 种组合。

如果骰子点数为 $6$ ，牌可以是 $6$ ，有 $1$ 种组合。

组合总数为共有 $60$ 个等可能组合，所以概率是 $\dfrac{11}{60}$ 。 $3+2+1+3+1+1=11.$

所以正确答案是 C。

If the rolled number was $1,$ then the tile can be $1,4,9,$ yielding $3$ combinations.

If the rolled number was $2,$ then the tile can be $2,8,$ yielding $2$ combinations.

If the rolled number was $3,$ then the tile can be $3,$ yielding $1$ combination.

If the rolled number was $4,$ then the tile can be $1,4,9,$ yielding $3$ combinations.

If the rolled number was $5,$ then the tile can be $5,$ yielding $1$ combination.

If the rolled number was $6,$ then the tile can be $6,$ yielding $1$ combination.

The total number of combinations is $3+2+1+3+1+1=11.$ There are $60$ combinations each with equal likelihood, so the probability is $\dfrac{11}{60} .$

Thus, the answer is C .

2008 AMC 8 第 24 题

答案：C

难度评级：1630

解答：

其他年份的第 24 题