HMMT November

Consider the elements of the sets mod 4. Then we would need to have sets of the form $\{0,0,0\}$, $\{0,2,2\},\{0,1,3\},\{1,1,2\}$, or $\{2,3,3\}$. In the set $\{1,2, \ldots, 19\}$ there are four elements divisible by 4 and 5 elements congruent to each of $1,2,3 \bmod 4$. Hence the desired number is given by $$\binom{4}{3}+\binom{4}{1}\binom{5}{2}+\binom{4}{1}\binom{5}{1}\binom{5}{1}+\binom{5}{2}\binom{5}{1}+\binom{5}{1}\binom{5}{2}=244$$