5. Twenty people, including $A, B, C$, sit down at a round table at random. What is the probability that at least two of $A, B, C$ sit next to each other?
Answer: $\frac{17}{57}$.

Solution: Let's number the seats around the table clockwise from 1 to 20. We will consider any (unordered) set of three seat numbers where $A, B, C$ are sitting as an outcome, and a favorable outcome as an unordered set of three seat numbers where $A, B, C$ are sitting if the numbers are not consecutive (the neighbors of number 20 are 19 and 1).

The number of all outcomes is $\frac{20 \cdot 19 \cdot 18}{6}$, since the first seat can be chosen in 20 ways, the second in 19, and the third in 18, but we agreed to consider unordered sets, so we divide by the number of permutations of three elements $3!=6$.

For favorable situations, there are two possibilities.

1) All three are sitting next to each other, i.e., they occupy seats $(1,2,3),(2,3,4), \ldots,(20,1,2)$. Clearly, there are 20 such triplets.
2) Exactly two are sitting next to each other, i.e., they occupy seats (1,2), (2,3), ..., (20,1), a total of 20 pairs of seats. If $A$ and $B$, for example, occupy seats (6,7), then the third cannot occupy seats 5,6,7,8, but can occupy any of the remaining seats, of which there are 16. By the multiplication principle, the number of favorable outcomes in this case is $20 \cdot 16$, and the total number of favorable outcomes is $20 + 20 \cdot 16 = 20 \cdot 17$.

By the classical definition of probability, $p=\frac{20 \cdot 17 \cdot 6}{20 \cdot 19 \cdot 18}=\frac{17}{57}$.

Comment: A correct and justified solution - 7 points. The probability can be found in different ways; in this solution, points are distributed as follows: the outcome and favorable outcome are defined - 1 point, the number of all outcomes is correctly found - 2 points, the number of favorable outcomes is correctly found - 4 points. For a computational error with correct reasoning, reduce by one point. If the solution is not completed, 1-2 points are given for proving useful auxiliary statements.