## Sum of Three-Digit Integers

What is the sum of all the positive three-digit integers that can be formed from the digits $2,3,5,6$, and $7$? Note that the same digit can appear more than once.
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
We can make $5\times 5\times 5=125$ different three-digit integers from $2,3,5,6$, and $7$
$222\: 223\: 225\: 226\: 227\:\cdots$
Since each integer has $3$ digits, we have a total of $125\times 3=375$ digits which we are going to distribute evenly among $3$ columns: the $100$ column, the $10$ column, and the unit column. Thus, each column has $375/3=125$ digits.
If we divide the 125 digits evenly among the $2,3,5,6$, and $7$, each digit will appear $125/5=25$ times in each column.
Sum of each column equals
$25(2+3+5+6+7)=575$
When we allow for place value, the sum of the integers equals
$575(100)+575(10)+575=63825$

Answer: $63825$