SOLUTION: 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?

Question 1209585: 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?

Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20063) (Show Source): You can put this solution on YOUR website!

Round table problems always falsely assume that the round table is placed on a huge turntable. That's not the way it is in reality, but that's the way it is always assumed to be in all round table problems in all algebra books I've ever seen in all my 40+ years of teaching math. Thus for any seating arrangement, turning the turntable in any direction does not alter the seating arrangement. There are 10 people. [It makes no difference whether they are students, female teachers, or male teachers]. Choose 1 person to sit facing north. [Remember, it doesn't matter which person sits facing north at first, because the turntable could be turned so that anybody could face north. So even though that person could be picked to face north at first 10 ways, it does not matter which person you pick. So be sure not to think you could pick that person 10 ways. (If you did choose that person 10 ways, you'd then have to divide that 10 by the 10 ways to turn the turntable so that each of the 10 would have a chance to be facing north, so it's just 1 choice to choose somebody to sit facing north.)] Then seat the other 9 people in the other 9 chairs in 9! ways. Answer 9! = 362880 ways. Edwin

Answer by ikleyn(52873) (Show Source): You can put this solution on YOUR website!
.
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?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For standard circular permutations, the general statement is:

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

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).

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