Starting from a number written in the decimal system, let's calculate the sum of the different numbers that can be obtained by the permutations of its digits. For example, starting from 110, the sum $110+101+11=222$ is obtained. What is the smallest number from which, starting, the obtained sum is 4933284?
Solution. In our problem, the number we are looking for cannot be a four-digit number or smaller, because from such a number, by rearranging its digits, we can obtain at most 24 different numbers, all of which are less than 10,000. Thus, the sum of these numbers is less than 240,000. Therefore, let's examine the five-digit numbers.

If any digit in the number we are looking for appears at least three times, then the number of different numbers that can be obtained by permuting the digits is at most 20, and each of these numbers is less than 100,000. Thus, the sum of these numbers is less than 2,000,000. Therefore, this is also not possible.

It is also not possible for two digits to each appear exactly twice, because from such a number, we can obtain 30 different numbers, the sum of which is less than \(30 \cdot 100000 = 3000000\).

Therefore, if there is a five-digit number that meets the conditions, then either all its digits are different, or exactly two of its digits are the same.

First, let's consider the case where all digits are different. The number is of the form \(\overline{a b c d e}\) (where \(a, b, c, d\), and \(e\) are all different digits). In this case, each digit appears 24 times in each place value when the digits are permuted. Thus, the sum of the numbers obtained is:

\[
\begin{aligned}
& 24 \cdot 11111 \cdot a + 24 \cdot 11111 \cdot b + 24 \cdot 11111 \cdot c + 24 \cdot 11111 \cdot d + 24 \cdot 11111 \cdot e = \\
& = 24 \cdot 11111 \cdot (a + b + c + d + e) = 4933284
\end{aligned}
\]

from which \(2 \cdot (a + b + c + d + e) = 37\) follows. This is clearly not possible, since the left side is an even number, while the right side is odd.

Finally, let's see if there is a five-digit number where exactly two digits are the same and it meets the conditions. Among the permutations of the digits of such a number, there is, for example, one of the form \(\overline{a a b c d}\), where \(a, b, c\), and \(d\) are different digits. If we fix one of the \(a\) digits in any place value, then by permuting the other digits, we get 24 numbers, so the \(a\) digit appears 24 times in each place value. If we fix any other digit in any place value, then the other digits have 12 different permutations, so each digit different from \(a\) appears 12 times in each place value.

Thus, the sum of the numbers obtained by permuting the digits of the number \(\overline{a a b c d}\) is: \(24 \cdot 11111 \cdot a + 12 \cdot 11111 \cdot b + 12 \cdot 11111 \cdot c + 12 \cdot 11111 \cdot d = 4933284\). Therefore, \(2a + b + c + d = 37\). We can then find the smallest five-digit number that meets this condition by choosing the smallest possible digit for the first place value. This is possible if the other place values have the largest possible digits; according to this, the largest possible digits are: \(2 \cdot 9 + 8 + 7 = 33\), so the digit in the first place value must be 4.

Therefore, there is a five-digit number among the five-digit numbers that satisfies the conditions (and among the smaller numbers, there is not), so the smallest such five-digit number is the smallest number we are looking for: 47899.