You mean you want P(AB, BC, AC, ABC) ?
Hmm...this is 1 - P(A,B,C separate)
P(A,B,C separate) = 4*3*2/4*4*4 = 6/16 = 3/8
So P(AB, BC, AC, ABC) = 1-3/8 = 5/8
I don't know how you got 20 arrangements, there should be 64 (4*4*4), 4 different options to place each of A,B,C.