There are 24 unique permutations (4*3*2*1) of four numbers; you have your choice of 4 for the first digit, 3 for the second, 2 for the third.

Each number will be in each position (thousands, hundreds, tens, ones) six times. (Proof: There are 24 permutations, and nothing unique in the behavior of each number, so the likelihood of being in each position must be uniform. 24/4 = 6.)

So each number (w,x,y,z) will be in 6 thousands places, 6 hundreds, etc.; w contributes w * (6000 + 600 + 60 + 6)

Sum of all possible permutations of (wxyz) == (w + x + y + z) * 6666