Henry Ernest Dudeney/Puzzles and Curious Problems/90 - Summing the Digits/Solution

by : $90$

 * Summing the Digits
 * What is the sum of all the numbers that can be formed with all $9$ digits ($0$ excluded),
 * using each digit once and once only, in every number?

Solution

 * $201 \, 599 \, 999 \, 798 \, 400$

Proof
There are $9!$ permutations of the $9$ digits, that is: $362 \, 880$.

Of these permutations, each digit appears in each of the $9$ positions a total of $8!$ times each.

We note that $\ds \sum_{n \mathop = 1}^9 n = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$.

Let $S$ be the required sum.

We have: