How can we explain there is not exist $6$-digit self-descriptive number?
Andrew Mclaughlin 
a self-descriptive number is an integer $m$ that in a given base $b$ is $b$ digits long in which each digit $d$ at position n (the most significant digit being at position $0$ and the least significant at position $b - 1$) counts how many instances of digit $n$ are in $m$.
For example, in base $10$, the number $6210001000$ is self-descriptive because of the following reasons:
In base $10$, the number has $10$ digits, indicating its base;
It contains $6$ at position $0$, indicating that there are six $0$s in $6210001000;$
It contains $2$ at position $1$, indicating that there are two $1$s in $6210001000$;
It contains $1$ at position $2$, indicating that there is one $2$ in $6210001000$;
It contains $0$ at position $3$, indicating that there is no $3$ in $6210001000$;
It contains $0$ at position $4$, indicating that there is no $4$ in $6210001000$;
It contains $0$ at position $5$, indicating that there is no $5$ in $6210001000$;
It contains $1$ at position $6$, indicating that there is one $6$ in $6210001000$;
It contains $0$ at position $7$, indicating that there is no $7$ in $6210001000$;
It contains $0$ at position $8$, indicating that there is no $8$ in $6210001000$;
It contains $0$ at position $9$, indicating that there is no $9$ in $6210001000$.
There is not exist $6$-digit self-descriptive number. But I don't know how to explain it. Please give me some help.
$\endgroup$ 21 Answer
$\begingroup$Clearly the sum of the digits is the total count of digits, so must equal 6. There are only a few ways to write 6 as the sum of 6 numbers (ignoring the order):
600000 510000 420000 411000 330000 321000 311100 222000 221100 211110 111111
In each case it is easy to check that it is not possible to order the digits to make it self-descriptive, because the count of digits does not match. For example, 411000 contains a 4, so some digit value has to occur 4 times, but none of them do.
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