Question on reasoning . . .

Question

George is younger brother of Michael. Michael asked George a question of a seven digit number: - The number of zeroes in the number gives the first digit of the number. - The number of ones in the number gives the second digit of the number. - The third digit of the number is the number of twos in the number and so on. Can you help George guess this interesting number?

Answer 1

$$3211000$$ has 3 zeros, 2 ones, 1 two, and 1 three.

Answer 2

The number cannot have leading zeroes, because then we'd immediately have a contradiction.

The number cannot start with $7$ or more, because we don't have that many digits.

The number cannot start with $6$. All other digits must be zero, but the last digit has to be nonzero, a contradiction.

The number cannot start with $5$. We need five zeroes. If we make the second to last digit a $1$, then we must make the second digit a $1$, but now we have less than five zeroes, a contradiction.

The number cannot start with $4$. If we make the fifth digit a $1$, then we have to make the second digit a $1$. But now we have two $1$s, and need to change another digit, but we can't, because we need four zeroes.

What about $3$? Make the fourth digit $1$. We now have a $1$, so we make the second digit $1$. But now we have two $1$s. We can make the second digit a $2$, and increment the third digit so it's a $1$, and everything still works!

What about $1$ or $2$ at the start? This limits the number of zeros we can put in the number, and we find quickly that we run out of room if we fill in a lot of spaces with stuff other than zero.