There's several answers to this puzzle, actually. Here is just one of them for your perusal:
ABCD, BADC, BDAC, DBCA, DCBA, CDAB, CADB, ACBD, CABD, ACDB, ADCB, DABC, DBAC, BDCA, BCDA, CBAD, BCAD, CBDA, CDBA, DCAB, DACB, ADBC, ABDC, BACD
Golden_Scroll submitted an alternate solution:
ABCD, ABDC, BADC, BACD, CABD, CADB, DACB, DABC, ADBC, ADCB, BDCA, BDAC, CDAB, CDBA, DCBA, DCAB, ACDB, ACBD, BCAD, BCDA, CBDA, CBAD, DBAC, DBCA