5
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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?

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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

$$\binom{20}{4}\binom{16}{4}\binom{12}{4}\binom{8}{4}.$$

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1
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Another way to see that is to put $20$ diplomats in a line and split in $5$ groups of $4$ diplomats. First $4$ would be one group, next $4$ would be the next group and go on. Thinking like that we have to know in how many ways we can put those guys in a line. That is $20!$, but for the first group there are $4!$ permutations that give us the same group and that happen for each group. So the total is equal to:

$$\frac{20!}{4!4!4!4!4!}$$

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