How many ways can 20 different diplomats be assigned to 5 different continenets? Well, there are 20 diplomats, and $any$ of the 5 continents can be assigned to them, so this is $5 * 5 * 5 * 5 ... * 5 = 5^{20}$ (any of the five for the first diplomat, any of the five for the second,  etc...)
The main question here is:
What if each continent needs to have 4 diplomats each? How would I do this then?

There are $\binom{20}{4}$ ways to assign four diplomats to the first continent,  $\binom{16}{4}$ ways to assig four diplomats to the second continent and so on. Thus the answer is 
$$\binom{20}{4}\binom{16}{4}\binom{12}{4}\binom{8}{4}.$$
