Question 1209585
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The number of ways to seat 3 students, 2 male teachers, and 4 female teachers 
around a round table with 10 chairs is how much?
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For standard circular permutations, the general statement is:


<pre>
    +---------------------------------------------------------------------------------+
    |                If there are n distinguishable objects,                          |
    | then there are (n-1)! distinguishable circular permutations of these n objects. |
    +---------------------------------------------------------------------------------+
</pre>

Since you came with more complicated problem with a vacant chair, you should be just 
familiar with these traditional problems with no vacant chairs.


In this problem, we have 3 students + 2 male teachers + 4 female teachers.
They all are distinguishable objects. So, there are 3+2+4 = 9 distinguishable objects.

Plus to it, we have one chair, which is #10 distinguishable object.


Thus, the total of distinguishable objects in this problem is 10.
All 10 objects are distinguishable and there are no repeating undistinguishable objects.


Therefore, according to the general statement, in this problem there are (10-1)! = 9!
distinguishable circular permutations.


So, this given problem is similar/identical to just familiar to you other problems 
on circular permutations with distinguishable objects.


The difference appears when the number of undistinguishable objects in the problem
(like undistinguishable vacant chairs) is 2 or more.  Then the value of different 
circular arrangement (n-1)!, where n is the number of all objects, in total, should be 
divided by  k!,  where k is the number undistinguishable objects (repeating copies).