\[\boxed{\mathbf{344}\mathbf{.}}\]
\[1)\ \frac{3}{19} \cdot 6 = \frac{3 \cdot 6}{19} = \frac{18}{19}\]
\[2)\ 2 \cdot \frac{7}{11} = \frac{2 \cdot 7}{11} = \frac{14}{11} = 1\frac{3}{11}\]
\[3)\ 7 \cdot \frac{3}{49} = \frac{7 \cdot 3}{49} = \frac{3}{7}\]
\[4)\ 32 \cdot \frac{5}{8} = \frac{32 \cdot 5}{8} = 4 \cdot 5 = 20\]