2013/14 AHSMC Part 1 Problem 12

Problem 12 of 16

A very small cinema has only one row of five seats numbered $1$ to $5$ . Five movie goers arrive one at a time. Each takes a seat not next to any occupied seat if this is possible. If not, then any seat will do. The number of different orders in which the seats may be taken is

Show solution

Solution

The first two movie-goers to arrive may take the pair $(1,5)$ , $(1,4)$ , $(2,5)$ , $(1,3)$ , $(2,4)$ or $(3,5)$ of seats. In each case, there are $2! = 2$ ways for them to do so. If they take $(1,5)$ , $(1,3)$ or $(3,5)$ , then the third movie-goer has only one choice of seat. The remaining two seats may be occupied in $2! = 2$ ways. Otherwise, the last three movie-goers may take any vacant seats, and this can be done in $3! = 6$ ways. Hence the total number of orders in which the seats may be taken is $2 \times 3 \times (2 + 6) = 48$ . The answer is (C).

University of Alberta

Answer choices

Solution