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

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Puzzles and Curious Problems by Henry Ernest Dudeney: $89$

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:

\(\ds S\) \(=\) \(\ds 6 \times \paren {1 + 2 + 3 + 4}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 6 \times \paren {10 + 20 + 30 + 40}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 6 \times \paren {100 + 200 + 300 + 400}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 6 \times \paren {1000 + 2000 + 3000 + 4000}\)
\(\ds \) \(=\) \(\ds 6 \times \paren {10 + 100 + 1000 + 10 \, 000}\)
\(\ds \) \(=\) \(\ds 66 \, 660\)

$\blacksquare$


Sources

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