Question 1208263: In how many ways can 3 female teachers, 2 female students, and 2 male students be arranged around a round table such that there is a female student between each pair of female teachers?"
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52786)   (Show Source):
You can put this solution on YOUR website! .
In how many ways can 3 female teachers, 2 female students, and 2 male students
be arranged around a round table such that there is a female student
between each pair of female teachers?
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Wording is monstrous, but I will try to do my best.
I will use a symbol F as a placeholder for female teachers;
a symbol G (as girls) as a placeholder for female students;
a symbol B (as boys) as a placeholder for male students.
As problem is worded, the only way to satisfy requirements is to have a block
FGFGF at circular siting.
Also, I will use a symbol D for this block FGFGF, when I will consider this block as one clued object.
Then
- there are 3! = 6 ways/permutations for 3 distinguished female teachers (F) inside this block;
- there are 2! = 2 ways/permutations for 2 distinguished female students (G) inside this block;
- in circular sitting, we can assume that the block D = FGFGF is in position "North", or 12 o'clock.
Then in circular sitting, there is only one possible remaining configuration DGG, or FGFGFBB.
Finally, since 2 permutations are possible for two distinguishable boys, it gives an additional factor 2.
Now the number of all possible arrangements is 3! * 2! * 2! = 6*2*2 = 24. ANSWER
Solved.
In this my solution, I considered participating people as distinguishable individuals.
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Why did I say that wording is monstrous ?
Because on a circle, there is NO such notion and such conception as "between two points".
It is and it makes sense for points on a line, but not on a circle.
So, a reader more rigorous mathematically than me could advise you
to through this problem away or, at least, to edit (to write) it properly.
This writing is good for those readers what are able to read the thoughts of a writer
immediately from the writer's head or "between the lines", or "behind the lines", which is not Math.
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
It has to be like this:
2! ways to seat the MS's, the male students.
3! ways to seat the FT's, the female teachers.
2! ways to seat the FS's, the female students.
(2!)(3!)(2!) =(2)(6)(2) = 24 ways.
Edwin
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