Problem:
A small airplane has rows of seats with 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 adjacent seats in the same row for the couple?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
For simplicity, we can disregard the order in which passengers are seated, so we only consider the number of ways that the seats can be filled by passengers for the total number of possibilities, which is given by .
Out of these possibilities, we consider the number of ways where no adjacent seats are available, then subtract this from . This scenario can happen when two passengers are occupying the edge seats of one row or one passenger is seated in the middle seat of a row.
If no passenger is in the middle seat (all passengers are on the edge seats), we have way.
If exactly passenger is in the middle seat (thus are seated on the edge seats), there are ways to choose the row with the middle seat occupied, and ways for the remaining passenger to be placed in the same row. So this gives ways.
If exactly rows have passengers in the middle seat, we have ways to choose the rows and ways to place the other passengers in those same rows, yielding ways.
If exactly rows have their middle seat occupied, there are ways to choose the rows and ways to seat the other passengers, giving ways.
If all rows have their middle seats occupied, the remaining passengers can be seated in ways.
Adding all the possibilities, we get ways for there to be no two adjacent seats available.
Hence, the probability that the couple will be seated together is
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions