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Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity:
$$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$
for all natural ...
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