The simplified value of $$  \left\{1\frac{1}{4}  of  \left(2\frac{1}{3} \div 1\frac{2}{5}\right) - 1\frac{5}{12}\right\} + \frac{1}{9} \div 2\frac{1}{3} + \frac{2}{7} + \frac{1}{6}$$ is:

$$  \left\{1\frac{1}{4}  of  \left(2\frac{1}{3} \div 1\frac{2}{5}\right) - 1\frac{5}{12}\right\} + \frac{1}{9} \div 2\frac{1}{3} + \frac{2}{7} + \frac{1}{6}$$

$$\left\{\frac{5}{4}\times\left(\frac{7}{3}\times\frac{5}{7}\right)-\frac{17}{12}\right\}+\frac{1}{9}\times\frac{3}{7}+\frac{2}{7}+\frac{1}{6}$$

$$\left(\frac{5}{4}\times\frac{5}{3}-\frac{17}{12}\right)+\frac{1}{21}+\frac{19}{42}$$

$$\frac{2}{3}+\frac{1}{21}+\frac{19}{42}=\frac{7}{6}$$