Question 1207620: At a meeting, $4$ scientists, $3$ mathematicians, and $2$ journalists are to be seated around a circular table. How many different arrangements are possible if every mathematician must sit next to a journalist? (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52798)   (Show Source):
You can put this solution on YOUR website! .
At a meeting, 4 scientists, 3 mathematicians, and 2 journalists are to be seated
around a circular table. How many different arrangements are possible if every mathematician
must sit next to a journalist? (Two seatings are considered equivalent if one seating
can be obtained from rotating the other.)
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Draw a circle - it will represent the circular table.
There are 9 chairs around the table.
Let assume that the chairs are numbered from 1 to 9 sequentially clockwise around the table
and let assume that the chair #1 is in position "North", or 12 o'clock.
We will place one of the two journalist at the chair #1.
Then the other journalist can not occupy neither of the two neighbor chairs,
since otherwise will be no place for 3 mathematicians next to two journalists.
It means that the other journalist can occupy any one chair from #3 to #8 inclusive.
Thus, there are 9 - 3 = 6 possibilities for the other journalist's chair.
OK. So, there are 6 possibilities to place two journalists. (1)
Next, assume that two journalists are just placed this way.
Then there are 4 chairs neighbor these two journalists chairs to place 3 mathematician there.
There are  = 4 ways to place 3 mathematicians at these 4 chairs. (2)
So, now we have 2+3 = 5 chairs occupied and 9-5 = 4 chairs free for four scientists.
These scientists can be placed in 4! = 24 different ways in these 4 chairs. (3)
Now calculate the product of options
n = 6 (from (1)) * 4 (from (2)) * 24 (from (3)) = 6 * 4 * 24 = 576.
At this point, the problem is solved to the end, and the number of
all different arrangements is 576. ANSWER
Solved.
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It is very nice combinatorics problem of an Olympiad level.
I never saw and never solved similar combinatorics problems before.
It is very rare case to see a new, a fresh and so beautiful combinatorics problem (!)
/ / / / / / / / / / / / / / / / / /
I agree with Edwin, noticing the missed factor 2.
So, the correct answer is 2*576 = 1152.
Thank you, Edwin !
Answer by Edwin McCravy(20059)  (Show Source):
You can put this solution on YOUR website!
Ikleyn has analyzed the logic of the problem correctly, and I applaud her for
that. But unfortunately, she overlooked the very small fact that there are 2
choices for the journalist to place in chair #1 in position "North", or 12
o'clock. So she only got half the correct answer. Since there are 2 ways to
choose the journalist in chair #1 at 12 o'clock, the correct answer is 2x576 or
1152.
Edwin
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