Adding a Million Numbers at Once

What is the sum of all the different nine-digit numbers that can be made by using each of the digits $1$ to $9$ exactly once?

Solution

There are $9! = 362880$ such numbers, and no one would add them one by one. By symmetry each digit spends equal time in each of the nine places: it sits in the units place in $8! = 40320$ of the arrangements, in the tens place in another $40320$, and so on. So each column, units through hundred-millions, has the same total, $$(1 + 2 + \dots + 9) \times 40320 = 45 \times 40320 = 1814400.$$ The whole sum is that column-total written into every place at once, that is multiplied by $111111111$: $$1814400 \times 111111111 = 201{,}599{,}999{,}798{,}400.$$