Summing 4-Digit Numbers with 1,2,3,4

[The legend]

Find sum!

Homework Statement

Find sum of all 4-digit numbers formed by digits 1,2,3,4 with repetition.

The Attempt at a Solution

i did come figure out that there are 4⁴ = 256 numbers possible but how do i sum them up without listing each individually?

 

[ehild]

Can you arrange them in increasing order? Which one is the smallest, the next and so on?

 

[The legend]

ok, that would give me something like this

1111
1112
1113
1114
1121
1122
.... and so on but how will that help?

 

[G037H3]

why don't you look at the vertical pattern of the numbers you're writing out...

 

[The legend]

ow foolish me!
so i get it there's repetition of 1+2+3+4 in units and in tens 1+1+1+1+2+2+2+2... so on and adding them a bit cleverly(though it does consume some time!) as you guys suggested and I got the answer.
so thanks for helping me out!

 

[ehild]

I would make an other column beside this one, but the numbers would be in decreasing order...

ehild

 

[The legend]

oh .. i see.. this is even quite easier than i suspected!

Thanks again!

 

[zgozvrm]

... so what did you come up with?

 

[ehild]

I would make an other column beside this one, but the numbers would be in decreasing order...

ehild

and I would add up the two numbers in one row...

ehild

 

[The legend]

adding up as ehild said i get same terms ie= 5555 256 times!
so what would happen is we get twice the sum actually needed and dividing it by 2 ... easy answer!