\[27 \cdot \left( \frac{2}{3} \right)^{2} \cdot 1\frac{7}{8} = 27 \cdot \frac{4}{9} \cdot \frac{15}{8} =\]
\[= \frac{3 \cdot 1 \cdot 5}{2} = \frac{15}{2} = 7\frac{1}{2}.\]