Shapovalov A.V.

At a round table, 12 places were prepared for the jury with each place labeled with a name. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his own place, but in the next one clockwise. Each subsequent jury member, approaching the table, would take their own place or, if it was already occupied, walk around the table clockwise and sit in the first available place. The resulting seating arrangement of the jury members depends on the order in which they approached the table. How many different seating arrangements can arise?
Let's consider a certain way of seating the jury members. We will call a jury member lucky if they are sitting in their own seat. The first of the unlucky ones (excluding Nikolai Nikolaevich) to approach the table is the one whose seat is taken by Nikolai Nikolaevich (another unlucky one would sit in their still free seat, which contradicts their unluckiness). He takes the seat of the next (clockwise) unlucky jury member. The second of the unlucky ones to approach is the one whose seat is taken by the first unlucky one (for the same reason), and so on. Thus, each unlucky one sits in the next "unlucky" seat after their own.

Therefore, the seating arrangement is uniquely determined by the division of the jury into lucky and unlucky members. Nikolai Nikolaevich and the one whose seat he took are always unlucky. Any subset of the jury members, not including these two, can be the set of lucky ones. Such a seating arrangement can be implemented, for example, as follows: after Nikolai Nikolaevich, all those we have chosen as lucky (in any order) enter, and then all the others in the order of their seating around the table clockwise. Therefore, the number of seating arrangements is equal to the number of subsets of a set of 10 people, which is \(2^{10}=1024\).

## Answer

1024 ways.