imc

We can figure out $H=0$ by noticing that $19!$ will end with $3$ zeroes, as there are three factors of $5$ in its prime factorization, so there would be 3 powers of 10 meaning it will end in 3 zeros. Next, we use the fact that $19!$ is a multiple of both $11$ and $9$. Their divisibility rules (see Solution 2) tell us that $T+M\equiv 3(\mathrm{mod}9)$ and that $T-M\equiv 7(\mathrm{mod}11)$. By guess and checking, we see that $T=4,M=8$ is a valid solution. Therefore the answer is $4+8+0=\boxed{12}$.