Solution — 2024 Question 7 Permutations & Combinations

(a)

\begin{matrix} Number of ways \\ = (\binom{15}{9}) 9! \\ = 1 816 214 400 ∎ \end{matrix}

(b)

We consider two cases based on where Gordon sits.

Case 1: Gordon sits at either end of the row.

There are 2 choices for Gordon’s seat. Only 1 adjacent seat is blocked, leaving 13 seats for the remaining 8 passengers.

Case 2: Gordon sits in one of the 13 middle seats.

Both adjacent seats are blocked, leaving 12 seats for the remaining 8 passengers.

\begin{matrix} Required number of ways \\ = 2 (\binom{13}{8}) 8! + 13 (\binom{12}{8}) 8! \\ = 363 242 880 ∎ \end{matrix}