AMC 12 2002b Problem 10

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

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