1. Stating the problem: Calculate $\frac{1}{3} \times \left( \frac{9}{3} + \frac{5}{6} \right)$.\n\n2. Use the order of operations: first solve inside the parentheses, then multiply.\n\n3. Simplify inside the parentheses:\n$$\frac{9}{3} = 3$$\nSo, inside parentheses becomes $$3 + \frac{5}{6}$$.\n\n4. Convert 3 to a fraction with denominator 6 to add easily:\n$$3 = \frac{18}{6}$$\nSo, $$\frac{18}{6} + \frac{5}{6} = \frac{18+5}{6} = \frac{23}{6}$$.\n\n5. Now multiply by $\frac{1}{3}$:\n$$\frac{1}{3} \times \frac{23}{6} = \frac{1 \times 23}{3 \times 6} = \frac{23}{18}$$.\n\n6. The fraction $\frac{23}{18}$ is an improper fraction and cannot be simplified further.\n\n7. Check the options: none match $\frac{23}{18}$, so re-check the problem. The problem asks for $\frac{1}{3} \times \left( \frac{9}{3} + \frac{5}{6} \right)$, which we solved correctly. The options given are different fractions, so the correct answer is not listed.\n\nFinal answer: $\frac{23}{18}$ (not among options).