Question

Let S be the set of distinct ordered triples comprised of the numbers 1, 2, 3, 4. To say that the triple is distinct means that no number occurs twice in the triple. To say that the triple is ordered means that two triples in which the same numbers appear in a different order are considered to be different triples. Some of the elements of S are: 1,2,3), (1,2,4), (3,2,1), (3,2,4), (4,2,1), (4,3,2) We wish to list all of the elements of S by a certain system. In this system the triple listed following (i, j, k) is one of the triples that begins with (j, k). For example, assuming that we start with (1,2,3), then the next triple listed must be either (2,3,1) or (2,3,4) Starting with (1,2,3), there are four possible ways to list the first three triples: The question is: can we list all 24 triples of S according to this system, starting with (1,2,3) and listing no triple twice, so that the last triple listed is a predecessor of (1,2, 3), or, in other words, is either (3,1,2) or (4,1,2)?'

Answer 1

3 1,2,3) (2 124 (4,(4,), (1A,), (314, (4,) 5,411 1,3 C1, 3,4), (3,4,1). C4,31), (1,4,3), (31,4)