2007 AMC 12A Problem 22

For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n + S(n) + S(S(n)) = 2007?

\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.