The 4! = 24 distinct fractions whose base 4 representation consists of permutations of the digits (0, look 1,2,3) repeated:

These are all fractions of the form , namely those fractions  with numerators 9, 10, 13, 15, 18, 19, 25, 26, 33, 36 ,38, 40, 45, 47 ,49 ,52, 59 ,60 66, 67, 70, 72, 75 and 76.