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

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About mvtrinh

Retired high school math teacher.
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