1. **Problem:** Calculate the product $(-\frac{1}{2}) \cdot (+\frac{2}{3}) \cdot (-\frac{3}{4}) \cdot (-\frac{4}{5}) \cdot (-\frac{5}{6})$. 2. **Formula and rules:** When multiplying fractions, multiply numerators together and denominators together: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$ Also, multiply the signs: an even number of negative factors results in a positive product, an odd number results in a negative product. 3. **Step-by-step calculation:** - Multiply numerators: $(-1) \times 2 \times (-3) \times (-4) \times (-5) = (-1) \times 2 = -2$, then $-2 \times (-3) = 6$, then $6 \times (-4) = -24$, then $-24 \times (-5) = 120$. - Multiply denominators: $2 \times 3 \times 4 \times 5 \times 6 = 720$. 4. **Combine:** $$\frac{120}{720}$$ 5. **Simplify:** Divide numerator and denominator by 120: $$\frac{120 \div 120}{720 \div 120} = \frac{1}{6}$$ 6. **Sign check:** There are four negative factors, which is even, so the product is positive. **Final answer:** $$\boxed{\frac{1}{6}}$$