## 23567

Find the sum of all positive three-digit integers formed from the five digits: $2, 3, 5, 6$, and $7$. The same digit can appear more than once.
Source: mathcontest.olemiss.edu 3/4/2013

SOLUTION
Let’s try a smaller problem to get an idea of what’s going on: sum of all 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 $9\left (11\right )$ to the sum.
The addition $20+30+60=110$ appears $9$ times contributing $9\left (110\right )$ to the sum.
The addition $200+300+600=1100$ appears $9$ times contributing $9\left (1100\right )$ to the sum.
Sum of all positive integers formed from three digits: $2, 3$, and $6$
$9\left (11\right )+9\left (110\right )+9\left (1100\right )=99+990+9900$
$=10989$
Now apply the pattern to our problem: sum of all positive integers formed from the five digits: $2, 3, 5, 6$, and $7$.
Total count of integers
$5\times 5\times 5=125$
The addition $2+3+5+6+7=23$ appears $25$ times contributing $25\left (23\right )$ to the sum.
The addition $20+30+50+60+70=230$ appears $25$ times contributing $25\left (230\right )$ to the sum.
The addition $200+300+500+600+700=2300$ appears $25$ times contributing $25\left (2300\right )$ to the sum.
Sum of all positive integers formed from the five digits: $2, 3, 5, 6$, and $7$
$25\left (23\right )+25\left (230\right )+25\left (2300\right )=575+5750+57500$
$=63825$

Answer: $63825$

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , . Bookmark the permalink.

### One Response to 23567

1. Dear Mr. Trinh,
I have found your math blog, which I really like it. I enjoyed to read your posts.
I was wondering if you would be interested in sharing your posts and ideas on Glipho? It’s a quite new social publishing platform, where you can connect to every social network accounts (Facebook, Twitter, Instagram, YouTube, etc). Really easy to use it and communicate with your followers. In additional, you are able to import your posts too.
Please, have a look and take a tour to know more about Glipho.
If you would like to set up your account, please do not hesitate to ask me for further information.
We are always delighted to get any feedback about the things which do or don’t work for you.
I hope you will join to Glipho community soon. It’s so much fun! 🙂
All the best,
Monika