2024 AMC 8 第 25 题

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

小提示：

前八名乘客坐下后，数剩下的四个空座

Count the four empty seats after the first eight passengers board

大提示：

用补集：没有任何一排有两个相邻空座

Use the complement: no row has two adjacent empty seats

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，即没有任何一排有两个相邻空座。在一排三个座位中，不含相邻空座的方案所含空座数可以是

0

、

1

或

2

，对应的方案数分别是

1

、

3

和

1

。因此，我们需要求下式中

x^4

的系数：

(1+3x+x^2)^4\text{。}

这个系数是

\begin{gathered} 3^4+4\cdot3\cdot3^2+\binom42 \\ =81+108+6 \\ =195 \end{gathered}\text{。}

因此至少有一排存在相邻空座的概率为

\frac{495-195}{495}=\frac{20}{33}\text{。}

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,

and

2,

with

1,

3,

and

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.