1. The council of the secret pipeline village is gathering around a round table, where each arriving member can sit in any free seat. How many different seating arrangements are possible if 7 participants will attend the council? (Two seating arrangements are considered the same if the same people sit to the left and right of each participant, empty seats are not considered.)
# Solution

Since empty seats are not taken into account, we can consider only the ways of arranging on seven seats. The first person can sit in any of the 7 seats, the next in any of the 6 remaining seats, and so on until the last. In total, there are $7 \cdot 6 \cdot \ldots \cdot 1=7!$ ways.

However, each arrangement that can be obtained from another by a cyclic shift (rotation clockwise or counterclockwise) is considered the same. For each fixed arrangement, there will be 7 such variants. Therefore, the final number is

$$
7 \cdot 6 \cdot \ldots \cdot 1 / 7=6!=720
$$

It is also worth noting that a mirror reflection transforms one seating arrangement into another, as the right and left neighbors switch places.

Answer: $6!=720$ ways.