## Task 13/84

Calculate the sum of all those natural numbers that contain each of the five digits $1 ; 2 ; 3 ; 4 ; 5$ exactly once in their decimal representation.
Since in the decimal representation of the numbers to be summed, each of the five numbers $1, 2, 3, 4, 5$ appears exactly once, there are $n=5!=120$ such numbers. If one imagines these numbers written one below the other, each of these digits appears in each column $\frac{5!}{5}=4!=24$ times. Consequently, for the sum $S_{i}$ of each column,

$$
S_{i}=24 \sum_{k=1}^{5} k=24 \cdot 15=360
$$

and for the desired sum $S$

$$
S=\sum_{i=1}^{5} S_{i} \cdot 10^{i-1}=360 \sum_{i=1}^{5} 10^{i-1}=360 \frac{10^{5}-1}{10-1}=3999960
$$