1. **State the problem:** Simplify the expression $\frac{1}{3}\left(\frac{1}{4}+\frac{1}{5}-\frac{1}{6}\right)$.\n\n2. **Use the formula:** We need to first simplify inside the parentheses by finding a common denominator and then multiply by $\frac{1}{3}$.\n\n3. **Find common denominator inside parentheses:** The denominators are 4, 5, and 6. The least common denominator (LCD) is 60.\n\n4. **Rewrite each fraction with denominator 60:**\n$$\frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60}$$\n\n5. **Perform addition and subtraction inside parentheses:**\n$$\frac{15}{60} + \frac{12}{60} - \frac{10}{60} = \frac{15 + 12 - 10}{60} = \frac{17}{60}$$\n\n6. **Multiply by $\frac{1}{3}$:**\n$$\frac{1}{3} \times \frac{17}{60} = \frac{1 \times 17}{3 \times 60} = \frac{17}{180}$$\n\n7. **Simplify fraction if possible:** 17 and 180 have no common factors other than 1, so the fraction is already in simplest form.\n\n**Final answer:** $$\frac{17}{180}$$