, 2013

Solution:

Answer: 66071772829247409

The sum is extremely unimodal, so we want to approximate it using its largest term. Taking logs of each term, we see that the max occurs when $(26-n) \log n$ peaks, and taking derivatives gives
$$x + x \log x = 26$$
From here it's easy to see that the answer is around $10$, and slightly less (it's actually about $8.3$, but in any case it's hard to find powers of anything except $10$). Thus the largest term will be something like $10^{16}$, which is already an order of magnitude within the desired answer $6.6 \times 10^{16}$.

To do better we'd really need to understand the behavior of the function $x^{26-x}$, but what approximately happens is that only the four or five largest terms in the sum are of any substantial size; thus it is reasonable here to pick some constant from $4$ to $20$ to multiply our guess $10^{16}$; any guess between $4.0 \times 10^{16}$ and $2.0 \times 10^{17}$ is reasonable.