smc

Define $T$ to be the set of all integral triples $(a,b,c)$ such that $a\ge b\ge c$, $b+c>a$, and $a,b,c<5$. Now we enumerate the elements of $T$: $(4,4,4)$ $(4,4,3)$ $(4,4,2)$ $(4,4,1)$ $(4,3,3)$ $(4,3,2)$ $(3,3,3)$ $(3,3,2)$ $(3,3,1)$ $(3,2,2)$ $(2,2,2)$ $(2,2,1)$ $(1,1,1)$ It should be clear that $|S|$ is simply $|T|$ minus the larger "duplicates" (e.g. $(2,2,2)$ is a larger duplicate of $(1,1,1)$). Since $|T|$ is $13$ and the number of higher duplicates is $4$, the answer is $13-4$ or $\boxed{9}$.