An international meeting is held between England, Germany, and France.  Three representatives attend from England, four from Germany, and two from France.  How many ways can all nine representatives sit around a circular table, if representatives of the same country sit together?  (Two ways are considered the same if one can be rotated to produce the other.)
I need to find the number of ways to arrange three groups of people around a circular table, and then multiply by the number of ways to arrange the people within each group. To arrange three groups around a circular table, I can fix one group at a certain position, and then there are two choices for the order of the other two groups. So there are 2 ways to arrange three groups around a circular table. To arrange the people within each group, I can use the formula for permutations: n! for n people. So for the English group, there are 3! ways, for the German group, there are 4! ways, and for the French group, there are 2! ways. Multiplying these together, I get 2 * 3! * 4! * 2! = 2 * 6 * 24 * 2 = 576 ways. 