Number of Solutions in this Number Puzzle

Question

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?

Answer 1

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.

Answer 2

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

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

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