There are 5! or 120 possible numbers.

if you were to list them all in a column to add, you'd have

  12345
  12354
  12435
  12453
  13245
  .....
  .....
  .....
  54213  
  54231  
  54312  
  54321
-------


Among all the 120 units digits, 1/5 or 24 of them are 1's, 
1/5 or 24 of them are 2's, ... , 1/5 or 24 of them are 5's

24 of them are 1's, so their sum is 24x1 =  24
24 of them are 2's, so their sum is 24x2 =  48
24 of them are 3's, so their sum is 24X3 =  72
24 of them are 4's, so their sum is 24x4 =  96
24 of them are 5's, so their sum is 24x5 = 120
----------------------------------------------
total of digits in the units column      = 360                           

So the sum of the digits in the units column is 360.  The other four
columns also have the same digits as the units digits.

So the sum of each column is 360.  So the sum is

360*10000 + 360*1000 + 360*100 + 360*10 + 360

360(10000 + 1000 + 100 + 10 + 1)

360(11111)

3999960

Edwin