There are 4! or 24 such numbers. So there are 24 digits in each column.
There are the same number of 1's in each column as there are 2's, 3's 
and 4's. so there are 24÷4 or 6 of each digit in each column.  So each
column sums to 6*1+6*2+6*3+6*4 = 6(1+2+3+4) = 6(10) = 60, so the sum is

the sum of the thousands column, which is 60, times 1000 plus 
the sum of the hundreds column, which is 60, times 100 plus 
the sum of the tens column, which is 60, times 10 plus
the sum of the units column, which is 60. 

1000(60) + 100(60) + 10(60) + 60 = 60(1000+100+10+1) = 60(1111)= 66660 

And since there are only 24 of them, you can check just by adding;


 1234
 1243
 1324
 1342
 1423
 1432
 2134
 2143
 2314
 2341
 2413
 2431
 3124
 3142
 3214
 3241
 3412
 3421
 4123
 4132
 4213
 4231
 4312
+4321
-----
66660

Edwin