Math Gold Medalist

Lor

2024 AMC 8 

Problem 25

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?

[DIAGRAM]

$\textbf{(A)}\ \dfrac{8}{15} \qquad \textbf{(B)}\ \dfrac{32}{55} \qquad \textbf{(C)}\ \dfrac{20}{33} \qquad \textbf{(D)}\ \dfrac{34}{55} \qquad \textbf{(E)}\ \dfrac{8}{11}$

Casework

Complement Rule

If we consider bad cases:

Number of passengers in each row should be 3,2,1(0 is not possible because we are considering bad cases and we shouldn’t have 2 empty adjacent seats in a row for the couple)

Cases of number of passengers in the rows:

Case1)3,3,1,1

Case2)3,2,2,1

Case3)2,2,2,2

.

   Solution