SOLUTION: In how many ways can 3 Americans, 4 Frenchmen, 4 Danes and 2 Italians be seated in a round table if those of the same nationality sit together?

We have 4 blocks of people according their nationalities.


4 items (4 blocks, in this case) can be placed around a circle in  (4-1)! = 3! = 1*2*3 = 6 different ways.


(Using mathematical language, for n different objects, there are (n-1)! different circular permutations).


Next, there are 3! =  6 permutations inside the block of Americans,
                4! = 24 permutations inside the block of Frenchmen,
                4! = 24 permutations inside the block of Danes, and
                2! =  2 permutations inside the block of Italians.


So, there are 6*(6*24*24*2) = 41472 different placements under imposed conditions.    ANSWER