I was studying for my combinatorics exam and one thing struck me;

How many ways are there to assign $100$ different diplomats to $5$ different continents?

So, I initially thought that the answer was $P(100,5)$, but the answer's actually $5^{100}$.

If I use $n^r$, the distribution should be unlimited. However, if I want to find the number of ways to assign $100$ individuals to $5$ different places, shouldn't it be considered "distribution with no repetition" thus $P(n,r)$, not $n^r$? Once a man is assigned to a place, he can't be assigned to other places at the same time, so $5^{100}$ doesn't really make sense? You assign a person in the continent A, there are $100$ options, in the continent B, there should be $99$ options left and so forth which leads to $P(100,5)$.

What am I missing here?


You let the first person choose which continet he wants to go to. He has $5$ options (continent to choose from).

After which, you let the second person choose which continent he wants to go to. He again, still has $5$ options (again, choosing continent).

Up to the $100$-th diploments, he still has $5$ options.

We assume each diplomat must be assigned somwhere but not every continent must have a diplomat.


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