## 23569 Sum

Using the five digits $2,3,5,6$, and $9$, what is the sum of all positive three-digit integers using these five digits? Please note that the same digit can appear more than once.
Source: mathcontest.olemiss.edu 4/15/2013

SOLUTION
This problem is similar to the problem titled “23567” dated 3/4/2013.
Let’s try a smaller problem to get an idea of what’s going on: sum of all three-digit positive integers formed from three digits: $2,3$, and $6$.
Total count of integers
$3\times 3\times 3=27$
$222\quad 322\quad 622$
$223\quad 323\quad 623$
$226\quad 326\quad 626$
$232\quad 332\quad 632$
$233\quad 333\quad 633$
$236\quad 336\quad 636$
$262\quad 362\quad 662$
$263\quad 363\quad 663$
$266\quad 366\quad 666$
The addition $2+3+6=11$ appears $9$ times contributing $9(11)$ to the sum.
The addition $20+30+60=110$ appears $9$ times contributing $9(110)$ to the sum.
The addition $200+300+600=1100$ appears $9$ times contributing $9(1100)$ to the sum.
Sum of all three-digit positive integers formed from three digits: $2,3$, and $6$
$9(11)+9(110)+9(1100)=99+990+9900$
$=10989$
Now apply the pattern to our problem: sum of all three-digit positive integers formed from the five digits: $2,3,5,6$, and $9$.
Total count of integers
$5\times 5\times 5=125$
The addition $2+3+5+6+9=25$ appears $25$ times contributing $25(25)$ to the sum.
The addition $20+30+50+60+90=250$ appears $25$ times contributing $25(250)$ to the sum.
The addition $200+300+500+600+900=2500$ appears $25$ times contributing $25(2500)$ to the sum.
Sum of all three-digit positive integers formed from the five digits: $2,3,5,6$, and $9$
$25(25)+25(250)+25(2500)=625+6250+62500$
$=69375$

Answer: $69375$