Evaluate the expression: 4 1/4 - 3 1/2 + 3 1/8 / (3 2/5 of 1 1/4 / 2 5/6)

For the numerator:

$$4\frac{1}{4} = \frac{4 \times 4 + 1}{4} = \frac{17}{4}$$ $$3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}$$ $$3\frac{1}{8} = \frac{3 \times 8 + 1}{8} = \frac{25}{8}$$

For the denominator:

$$3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$$ $$1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4}$$ $$2\frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{17}{6}$$

$$\frac{17}{4} - \frac{7}{2} + \frac{25}{8}$$

Find a common denominator, which is 8.

$$\frac{17 \times 2}{4 \times 2} - \frac{7 \times 4}{2 \times 4} + \frac{25}{8} = \frac{34}{8} - \frac{28}{8} + \frac{25}{8}$$ $$\frac{34 - 28 + 25}{8} = \frac{6 + 25}{8} = \frac{31}{8}$$

The expression is $3\frac{2}{5} \text{ of } 1\frac{1}{4} \div 2\frac{5}{6}$. "of" means multiplication.

$$\frac{17}{5} \times \frac{5}{4} \div \frac{17}{6}$$

First, perform the multiplication:

$$\frac{17}{5} \times \frac{5}{4} = \frac{17 \times 5}{5 \times 4} = \frac{17}{4}$$

Next, perform the division. To divide by a fraction, multiply by its reciprocal:

$$\frac{17}{4} \div \frac{17}{6} = \frac{17}{4} \times \frac{6}{17}$$ $$\frac{17 \times 6}{4 \times 17} = \frac{6}{4} = \frac{3}{2}$$

$$\frac{\frac{31}{8}}{\frac{3}{2}} = \frac{31}{8} \div \frac{3}{2}$$

Multiply by the reciprocal of the denominator:

$$\frac{31}{8} \times \frac{2}{3} = \frac{31 \times 2}{8 \times 3}$$

Simplify by canceling out a factor of 2:

$$\frac{31 \times 1}{4 \times 3} = \frac{31}{12}$$

Convert the improper fraction to a mixed number:

$$\frac{31}{12} = 2\frac{7}{12}$$

The final answer is $2\frac{7}{12}$.