smc

It is well-known that $n\equiv S(n)\equiv S(S(n))(\mathrm{mod}9).$ Substituting, we have that $$n+n+n\equiv 2007(\mathrm{mod}9)⟹n\equiv 0(\mathrm{mod}3).$$ Since $n\le 2007,$ we must have that $\max S(n)=1+9+9+9=28.$ Now, we list out the possible values for $S(n)$ in a table, noting that it is a multiple of $3$ because $n$ is a multiple of $3.$ Then, we compute the corresponding values of $S(S(n)).$ Finally, we may compute the corresponding values of $n$ using the fact that $n=2007-S(n)-S(S(n)).$ Notice how all conditions are designed to be satisfied except whether $S(n)$ is accurate with respect to $n.$ So, the only thing that remains is to check this. We may eliminate, for example, when $n=2007$ we have $S(n)=9$ while the table states that it is $0.$ Proceeding similarly, we obtain the following table. It follows that there are $\boxed{4}⟹\text{(D)}$ possible values for $n.$