2. (5 points) A company of 10 friends sits down at a round table in any order. Among them, there is one Vanya and one Dima. What is the probability that they will end up sitting next to each other?

## Solution:

Let's count the number of seating arrangements where Vanya and Dima are sitting next to each other. If Vanya has chosen a seat, Dima can choose one of the two adjacent seats. After their choice, there will be 8 seats left, which can be occupied by the remaining 8 people in 8! ways. Vanya can choose from 10 seats, after which Dima can choose one of 2 seats. In total, there are "favorable" seating arrangements of $10 \cdot 2 \cdot 8!$. The total number of possible seating arrangements is 10!. Therefore, the desired probability is $\frac{10 \cdot 2 \cdot 8!}{10!}=\frac{2}{9}$.
# Answer: $\frac{2}{9}$

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