Suppose you had them all lined up in a column to add:

  23578
  23587
  23758
  23785
  23857
  23875
  25378
   ...
  85732
  87235
  87253
  87325
  87352
  87523
 +87532
=======

There are 5!=120 numbers to be added.  One 5th of
the numbers in each column are 2's, one-fifth are 3's,
and so on to one-fifth are 8's,  So since one-fifth of
120 is 24, the sum of each column is 

24*(2+3+5+7+8) = 24*(25) = 600

So there are 

600 ten-thousands +
600 thousands +
600 hundreds +
600 tens +
600 ones

or

 6000000 +
  600000 +
   60000 +
    6000 +
     600 
========
 6666600

Edwin