Sum of the Numbers

There are 4! = 24 arrangements beginning with the number 1. Similarly there are 24 numbers beginning with each of 2 through to 5.

However, there are also 24 numbers with 1 as, say, the second digit. Similarly for any digit in any of 5 positions.

Hence, for example, the sum of all the units digits will be $24 \times 1 + 24 \times 2 + 24 \times 3+ 24 \times 4+ 24 \times 5 = 24(1+2+3+4+5) = 24 \times 15 = 360$ .

The same is true for other digits, so the sum required is 360 + 3600 + 36000 + 360000 + 3600000 = 3999960.

The total numbers can be formed by the five digits 1, 2, 3, 4 and 5 is $5! = 120$ . We note that each of the 1 to 5 digits must appear equal times, which means each digit appears $120 \div 5 = 24$ times each as the unit, ten, hundred, thousand and ten-thousand digits. Therefore, the sum of all the numbers formed by 1 to 5 digits is:

$\begin{aligned} S & = (1+2+3+4+5)(10^4+10^3+10^2+10^1+10^0)(24) = 15 \times 11111 \times 24 = \boxed{3999960} \end{aligned}$ .