+28 At a meeting, 4 mathematicians, 3 scientists, and 2 journalists are to be seated around a circular table. How many different arrangements are possible if the mathematicians must all sit together (in 4 consecutive seats) and the 3 scientists must sit together? (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)
+1839 We can think about this problem in two steps:
Step 1: Arrange the mathematicians and scientists as groups.
Since the mathematicians must sit together and the scientists must sit together, we can treat them as single units for now. This leaves us with 3 units (mathematicians, scientists, journalists).
There are 3! ways to arrange these 3 units around the circular table.
Step 2: Arrange the people within each group.
The mathematicians can arrange themselves in 4! ways within their designated section.
The scientists can arrange themselves in 3! ways within their designated section.
However, we've overcounted the arrangements because rotating the entire table doesn't create a new seating arrangement.
Step 3: Correct for overcounting due to circular symmetry.
Since there are 9 people sitting around the table, there are 9 ways to rotate the table so it appears as a new arrangement.
Total Arrangements:
To get the final number of arrangements, we take the number of arrangements from steps 1 and 2 and divide by the number of times we've overcounted in step 3:
Arrangements = (Arrangements of groups) * (Arrangements within groups) / (Overcounting)
Arrangements = (3!) * (4! * 3!) / (9)
Arrangements = 6 * 24 * 6 / 9
Arrangements = 864 / 9
Arrangements = 96
The answer is 96.
