2002 AMC 12B Problem 10

How many different integers can be expressed as the sum of three distinct members of the set \{1,4,7,10,13,16,19\}?

\mathrm{(A)}\ 13 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 24 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 35

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