# Math Help - 3-fold triples

1. ## 3-fold triples

A 3-fold truple system is an ordered pair (S,T). S is finite set and T is collection of 3-element sets called triples of S such that every pair of distinct elements of S appears in exactly 3 triples.

Show that a 3-fold triple system of order n(cardinality of S) exists if and only if n is odd.

2. I am having trouble coming up with a 3-fold triple system for $S=\{1,2,3\}$. The only possible triple seems to be $S$ itself, so it is not clear how the pair $\{1,2\}$ may be a subset (is this the meaning of "appears in"?) three different triples. Provided " $T$ is collection" means " $T$ is a set", so that, for example, a multiset $T=\{S, S, S\}$ is not allowed.