1. **Problem:** Evaluate the expression without using a calculator or mathematical table: $$\frac{5}{6} \text{ of } (4 \times 1 \times 3 \times 5 \times 5 \times 6) \times \frac{5}{6} \times \frac{5}{3} \times \frac{5}{2} \times \frac{12}{25} \times \frac{19}{3}$$ 2. **Step 1:** Calculate the product inside the parentheses: $$4 \times 1 = 4$$ $$4 \times 3 = 12$$ $$12 \times 5 = 60$$ $$60 \times 5 = 300$$ $$300 \times 6 = 1800$$ 3. **Step 2:** Multiply by $\frac{5}{6}$: $$\frac{5}{6} \times 1800 = 5 \times 300 = 1500$$ 4. **Step 3:** Multiply by the remaining fractions: $$1500 \times \frac{5}{6} = 1500 \times 0.8333 = 1250$$ $$1250 \times \frac{5}{3} = 1250 \times 1.6667 = 2083.33$$ $$2083.33 \times \frac{5}{2} = 2083.33 \times 2.5 = 5208.33$$ $$5208.33 \times \frac{12}{25} = 5208.33 \times 0.48 = 2500$$ $$2500 \times \frac{19}{3} = 2500 \times 6.3333 = 15833.33$$ 5. **Step 4:** Final answer rounded to nearest integer: $$\boxed{15833}$$ This is the evaluated value of the expression.