Question

一架小飞机有 $4$ 排座位，每排 $3$ 个座位。已有八名乘客登机，并随机分布在座位上。接下来一对夫妻登机。存在同一排中 $2$ 个相邻空座给这对夫妻的概率是多少？

A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be $2$ adjacent seats in the same row for the couple?

$\dfrac{8}{15}$

$\dfrac{32}{55}$

$\dfrac{20}{33}$

$\dfrac{34}{55}$

$\dfrac{8}{11}$

答案：C

知识点：基本概率补集计数生成函数

难度评级：1950

视频讲解：

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文字解答：

前 $8$ 名乘客坐下后，还有 $4$ 个空座分布在 $12$ 个座位中，所以空座集合共有 $\binom{12}{4}=495$ 种等可能情况。

用补集计数：没有任何一排有两个相邻空座。对一排三个座位，空座数为 $0,1,2$ 且不相邻时，选择数分别为 $1,3,1$ 。因此补集数是 $x^4$ 在中的系数： $(1+3x+x^2)^4.$ $\begin{gathered} 3^4+4\cdot3\cdot3^2+\binom42 \\ =81+108+6 \\ =195. \end{gathered}$

因此至少有一排存在相邻空座的概率为 $\frac{495-195}{495}=\frac{20}{33}.$

所以正确答案是 C。

After the first $8$ passengers sit, there are $4$ empty seats among the $12$ seats, so there are $\binom{12}{4}=495$ equally likely sets of empty seats.

Count the complement, where no row has two adjacent empty seats. In a row of three seats, the possible empty-seat patterns with no adjacent empty seats have sizes $0,1,2$ , with $1,3,1$ choices respectively. So we need the coefficient of $x^4$ in $(1+3x+x^2)^4.$ This coefficient is $\begin{gathered} 3^4+4\cdot3\cdot3^2+\binom42 \\ =81+108+6 \\ =195. \end{gathered}$

Therefore the probability that at least one row has adjacent empty seats is $\frac{495-195}{495}=\frac{20}{33}.$

Thus, C is the correct answer.

2024 AMC 8 第 25 题

答案：C

难度评级：1950

视频讲解：

文字解答：

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