Question

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.

Answer 1

Well, one obvious approach would be to list all of the possible numbers and add them up.

Answer 2

Or you could think about the problem as independent columns of digits. Let's look at just one column:

2
3
5
6
7

Those are all of the 1 digit numbers with those digits. The total is 2+3+5+6+7 = 23

What if we had two columns? Well, we would have 2 followed by all of those 1 digit numbers, then 3 followed by all of those 1 digit numbers, then 5 followed by all of those 1 digit numbers, then 6 followed by all of those 1 digit numbers, then 7 followed by all of those 1 digit numbers. Our sum now would be

2*10*5 + 2+3+5+6+7
3*10*5 + 2+3+5+6+7
5*10*5 + 2+3+5+6+7
6*10*5 + 2+3+5+6+7
7*10*5 + 2+3+5+6+7
---------
(2+3+5+6+7)10*5 + (2+3+5+6+7)5 = 5(10+1)(2+3+5+6+7) = 1265

Can you expand the pattern to have another column of digits?

Answer 3

For extra credit, what would be the sum of all 4 digit numbers drawn from digits 1,3,5,7?