1. Six people sit at a round table. in how many ways can that be done if two of them must sit next to each other?
To solve this problem, we can consider the two people who must sit next to each other as a single entity. This reduces the problem to arranging five entities (the group of two people and the remaining four individuals) at a round table.
First, let's calculate the total number of ways to seat six people at a round table without any restrictions. We can treat one person as a fixed reference point and arrange the other five people around them. This can be done in (5-1)! = 4! = 24 ways.
Now let's focus on the two people who must sit together. Since they are considered as a single entity, we can arrange them in 2! = 2 ways. However, since the round table does not have a distinct starting point, we need to divide this number by 6 to avoid overcounting (as we have six different starting points).
Therefore, the final answer is (4! * 2) / 6 = 8 ways to seat the six people at the round table if two individuals must sit next to each other.