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.)
To begin, consider the number of ways to arrange the three countries around the circle.  We can consider the English representatives a block, the Germans another block, and the French a third block.  There are $(3-1)!=2$ ways to arrange these three blocks around a circle.  We can also see this by simply drawing the two possible arrangements: [asy]
label("E",(0,0));
label("F",(-.75,-1));
label("G",(.75,-1));
label("E",(3,0));
label("F",(3.75,-1));
label("G",(2.25,-1));
[/asy] Within the English group, there are $3!=6$ ways to arrange the three representatives.  Similarly, there are $4!$ ways to arrange the Germans and $2!$ ways to arrange the French representatives.  Overall, the total number of ways to seat the 9 representatives is: $$2!\times3!\times4!\times2!=2\times6\times24\times2=\boxed{576}$$