6. Seven people seat themselves indiscriminately at a round table. The probability that two distinguished persons will be next to each other is A. \( 1 / 3 \) B. \( 1 / 2 \) C. \( 1 / 4 \) D. \( 2 / 3 \)

To find the probability that two distinguished persons will sit next to each other at a round table with seven people, we can solve it step by step. First, we can "fix" one of the distinguished persons in place due to the circular nature of the seating, which simplifies our arrangement. Imagine we have two distinguished persons, A and B. If we treat them as a single block (AB or BA), we now have 6 blocks to arrange (the AB block and the 5 other people). The total arrangements of these 6 blocks around the table is (6 - 1)! = 5! = 120. Now, A and B can switch within their block (AB or BA), giving us 2 configurations for them. Therefore, the number of ways in which A and B can sit together is 2 * 5! = 240. The total arrangements of all 7 people around a circular table is (7 - 1)! = 6! = 720. To find the probability that A and B sit next to each other, we divide the favorable outcomes by the total outcomes: \[ P(A \text{ next to } B) = \frac{240}{720} = \frac{1}{3} \] So, the correct answer is A. \( 1 / 3 \).