What are the $393$ trillion possible answers when computing $13^{th}$ root of a number? Alexis Lemaire got famous after finding the 13th root of a computer-generated 200-digit number without calculator. In this article, they say that there are "with 393 trillion possible answers to choose from". At first, I thought that it was the number of permutations of digits for each possible answer, but this wouldn't give the $393$ trillion, I guess. Does it actually mean anything or is it just a misunderstanding on the part of the journalists?
 A: This is just an elaboration on Raymond Manzoni's answer, addressing the OP's question whether the journalist misunderstood something.
Here are the lede paragraphs from the news story (which features a photo illustration of Lemaire pondering the 200-digit number):

When the answer is 2,407,899,893,032,210 you know the question is
  tough.
Not so tough, however, that Alexis Lemaire could not work it out in
  his head. His challenge yesterday was to come up with the 13th root of
  a computer-generated 200-digit number.
And, with 393 trillion possible answers to choose from, the PhD
  student made it almost look easy.

The "393 trillion" almost certainly came from a press release prepared by the sponsor of the challenge; no reporter (except maybe me) has the time or the expertise to make such a calculation (and I would probably get it wrong).  Given that the 13th root of the 200-digit number was an integer, it seems fairly clear that what the computer did was pick a (random?) number $n$ such that $10^{199}\le n^3\lt10^{200}$, so that there are $\lfloor10^{200/13}\rfloor-\lfloor10^{199/13}\rfloor\approx393$ trillion choices for $n$.
It's perhaps worth noting, though, that $a^{13}\equiv a$ mod $10$ for all $a$, which means that that a quick look at the final digit of the 200-digit number tells you the final digit of the answer.  So in a sense this quickly cuts the number of possible answers down to around $39.3$ trillion.  And in this case, in fact, the 200-digit number ends in a string of thirteen $0$'s preceded by a $1$, which means the answer necesarrily ends in $10$, thus cutting the number of possibilities to a "manageable" $3.9$ trillion....
