To solve this problem, let's break it down into steps:
Step 1: Consider the scientists as one group, since they need to sit together. We can think of them as a single entity. So, now we have 6 entities: the group of scientists, the two mathematicians, the journalist, and the remaining three seats around the table.
Step 2: Since the mathematicians must sit next to each other, we can think of them as a single entity as well. Now we have 5 entities: the group of scientists, the mathematicians, the journalist, and the remaining three seats.
Step 3: Let's arrange the 5 entities around the table. We can think of this as arranging the entities in a line, and then considering the starting point. There are 5! (5 factorial) ways to arrange the entities in a line.
Step 4: However, since the table is circular and rotating the seating arrangement doesn't create a new arrangement, we need to divide by 5 to account for the rotations. This is because if we fix one entity (say, the group of scientists) in a specific position, we can rotate the other entities around the table without changing the overall arrangement.
Step 5: Finally, the mathematicians can be arranged amongst themselves in 2! (2 factorial) ways. Step 6: Putting it all together, the total number of different arrangements is (5! / 5) * 2! = 4! * 2! = 24 * 2 = 48.
Therefore, there are 48 different arrangements possible if the scientists must all sit together and the mathematicians must sit next to each other.
- Jade!
