Henry Ernest Dudeney/Puzzles and Curious Problems/89 - Forming Whole Numbers/Solution

by : $89$

 * Forming Whole Numbers
 * Can the reader give the sum of all the whole numbers that can be formed with the four figures $1$, $2$, $3$, $4$?
 * That is, the addition of all such numbers as $1234$, $1423$, $4312$, etc.


 * You can, of course, write them all out and make the addition,
 * but the interest lies in finding a very simple rule for the sum of all the numbers that can be made with $4$ different digits selected in every possible way, but $0$ excluded.

Solution

 * $66 \, 660$

Proof
There are $24$ permutations of $1$, $2$, $3$ and $4$.

Of these permutations, each digit appears in each of the $4$ positions $6$ times each.

Let $S$ be the required sum.

We have: