SOLUTION: Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults ar

Question 1207574: Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other? (Two seatings are considered the same if one can be rotated to form the other.)
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
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Four children and four adults are to be seated at a circular table.
In how many different ways can they be seated if all the children are next to each other,
and all the adults are next to each other? (Two seatings are considered the same
if one can be rotated to form the other.)
~~~~~~~~~~~~~~~~~~~~

Again, as in one of problems that I solved for you earlier today, we subdivide 
the people in two separate groups. One group is of 4 children; the other group 
is of 4 adults.


First, we consider all possible circular permutations of these two groups as 
of two whole objects.
The number of all possible circular permutations of these two groups is

    (2-1)! = 1! = 1.


Next, there are 4! = 24 permutations inside the group of 4 children and 
another 4! = 24 permutations inside the group of 4 adults.
These permutations are independent and produce 24*24 = 576 permutations,
in total.


So, there are 576 circular permutations, in all.    ANSWER

Solved.

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