Combinatorics · Bijections · Involutions · Permutations · Processes
A cinema hall has \(2015\) seats, and \(2014\) viewers, one of whom is Mika, walk in. They all sit down on arbitrary seats, paying no attention to the seat assigned to them by their ticket, so exactly one seat stays empty.
Halfway through the film the \(2015\)th viewer enters. He insists on sitting exactly on the seat named by his ticket, and if that seat is taken he makes the viewer sitting there stand up. The viewer who has been made to stand then looks up his own seat, and if it is taken he in turn makes its occupant stand up. This continues until a viewer is made to stand whose own ticketed seat happens to be free, and he then sits down on it.
Let \(a\) be the number of initial seatings for which Mika is made to stand at some moment during all this commotion, and let \(b\) be the number of the remaining initial seatings. Which of the three relations
holds?
Sign in to check answers, open hints, read the full solution, and track your progress. Statements are always free.
Serbian Regional Competition (Okruzno takmicenje) 2015, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source