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At a meeting, five scientists, two mathematicians, and a journalist are to be seated around a circular table. How many different arrangements are possible if the scientists must all sit together (in five consecutive seats) and the mathematicians must sit next to each other? (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)

 Aug 20, 2023
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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!

 Aug 20, 2023

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