2024 AMC 8 Problem 25

Attempt Problem 25 of the 2024 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 8 solutions, or check the answer key.

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25.

A small airplane has 44 rows of seats with 33 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 22 adjacent seats in the same row for the couple?

815\dfrac{8}{15}

3255\dfrac{32}{55}

2033\dfrac{20}{33}

3455\dfrac{34}{55}

811\dfrac{8}{11}

Answer: C
Concepts:basic probabilitycomplementary countinggenerating functions
Difficulty rating: 1950
Video solution:
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Written solution:

After the first 88 passengers sit, there are 44 empty seats among the 1212 seats, so there are (124)=495\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 0,1,20,1,2, with 1,3,11,3,1 choices respectively. So we need the coefficient of x4x^4 in (1+3x+x2)4. (1+3x+x^2)^4. This coefficient is 34+4332+(42)=81+108+6=195. \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 495195495=2033. \frac{495-195}{495}=\frac{20}{33}.

Thus, C is the correct answer.

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