Question

一个盒子里有 $2$ 枚一美分硬币、 $4$ 枚五美分硬币和 $6$ 枚十美分硬币。在不放回且每枚硬币被选中的概率相同的情况下抽取六枚硬币。抽到的硬币总值至少为 $50$ 美分的概率是多少？

A box contains $2$ pennies, $4$ nickels and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of the coins drawn is at least $50$ cents?

$\frac{37}{924}$

$\frac{91}{924}$

$\frac{127}{924}$

$\frac{132}{924}$

以上都不是

none of these

答案：C

知识点：组合无放回抽样分类讨论

难度评级：2100

小提示：

共有 $\binom{12}{6}$ 种等可能的六枚硬币组合

There are $\binom{12}{6}$ equally likely sets of six coins

大提示：

按照抽到六枚、五枚或四枚十美分硬币分类计数有利情形

Classify favorable selections by whether they contain six, five, or four dimes

解答：

共有 $\binom{12}{6}=924$ 种选法。总值至少为 $50$ 美分的情形包括：六枚都是十美分硬币；五枚十美分硬币加任意一枚其他硬币；或四枚十美分硬币加两枚五美分硬币。有利选法数为 $\begin{aligned} &\binom66\\ &\quad+\binom65\binom61\\ &\quad+\binom64\binom42\\ &=1+36+90\\ &=127 \end{aligned}\text{。}$ 因此概率为 $\frac{127}{924}$ 。

因此，正确答案是 C。

There are $\binom{12}{6}=924$ selections. A value of at least $50$ cents occurs with six dimes; with five dimes and any one other coin; or with four dimes and two nickels. The favorable count is $\begin{aligned} &\binom66\\ &\quad+\binom65\binom61\\ &\quad+\binom64\binom42\\ &=1+36+90\\ &=127. \end{aligned}$ Thus the probability is $\frac{127}{924}.$

Therefore, the correct answer is C.

1980 AMC 12 第 20 题

答案：C

难度评级：2100

小提示：

大提示：

解答：

其他年份的第 20 题