Number of Solutions in this Number Puzzle

Today in math class, a number puzzle arose where you would have to find a 10 digit number, where each digit describes the number of other digits in the number, for example:
$$\text{0 1 2 3 4 5 6 7 8 9}$$ $$\text{6 2 1 0 0 0 1 0 0 0}$$

$\textit{The top row being digit number$n$, and the bottom row being the$10$digit number.}$

You can see that since there are six zeros in the $10$ digit number, there is a $6$ in the $0$'s column. And since there are two $1$'s, there is a $2$ in the $1$'s column and so on.

So, my question would be: Are there any more solutions to this puzzle? And if not, how could you prove this?

• A cute puzzle indeed! – Peter Sep 8 '15 at 22:05
• Must be the first row $0123456789$ or can it be something else? – ajotatxe Sep 8 '15 at 22:07
• @ajotatxe The first row is just describing the number of digits, except it starts at 0. But yes. – Jamie Sanborn Sep 8 '15 at 22:10
• this comment has 1 0, 11 1s, 2 2s, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, and 1 9. – Neil W Sep 9 '15 at 3:37

Let $k$ be the number of zeroes; clearly $k>0$, so there are $9-k$ other non-zero digits, and they must sum to $10-k$. That’s possible if and only if one of them is a $2$, and the rest are $1$’s. One of the $1$’s must represent the number of $k$’s, the other must represent the number of $2$’s, and the $2$ must therefore represent the number of $1$’s. Thus, the given solution is the only one.

• The number of non-zero digits must be $10-k$, if $k$ is the number of zeros. Or do I miss somthing ? – Peter Sep 8 '15 at 22:25
• @Peter: The number of other non-zero digits (i.e., besides the non-zero digit $k$) is $9-k$. – Brian M. Scott Sep 8 '15 at 22:28
• Ok, I got it now! – Peter Sep 8 '15 at 22:30

Here's a brute-force proof, just for fun.

Empirical confirmation of @BrianM.Scott's answer takes but a few minutes.

...
62.000% complete
62.100% complete
6210001000
62.200% complete
62.200% complete
...