6

Решение:

1. \( \frac{2}{5} \cdot \frac{1}{3} = \frac{2 \cdot 1}{5 \cdot 3} = \frac{2}{15} \)
2. \( \frac{2}{15} \cdot \frac{3}{4} = \frac{2 \cdot 3}{15 \cdot 4} = \frac{6}{60} = \frac{1}{10} \)
3. \( \frac{1}{10} \cdot \frac{5}{6} = \frac{1 \cdot 5}{10 \cdot 6} = \frac{5}{60} = \frac{1}{12} \)

Ответ: \( \frac{1}{12} \)