To solve the expression \((-2) \times \left(\frac{1}{5}\right) \div \left(\frac{2}{3}\right) \times \left(-\frac{5}{6}\right) \div \left(\frac{3}{4}\right)\), we can follow the order of operations and handle multiplication and division from left to right:
- Start with the first operation: \((-2) \times \left(\frac{1}{5}\right)\):
\[ -2 \times \frac{1}{5} = -\frac{2}{5} \]
- Now, we move to the next operation, which is division by \(\left(\frac{2}{3}\right)\):
\[ -\frac{2}{5} \div \frac{2}{3} = -\frac{2}{5} \times \frac{3}{2} = -\frac{2 \times 3}{5 \times 2} = -\frac{6}{10} = -\frac{3}{5} \]
- Next, we multiply by \(-\frac{5}{6}\):
\[ -\frac{3}{5} \times -\frac{5}{6} = \frac{3 \times 5}{5 \times 6} = \frac{15}{30} = \frac{1}{2} \]
- Finally, we divide by \(\left(\frac{3}{4}\right)\):
\[ \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
Thus, the final result of the expression is:
\[ \frac{2}{3} \]
The correct answer is C. \( \frac{2}{3} \).