To solve the given expression, let's evaluate it step by step. The expression is:

\frac{4}{5} + \left(\frac{1}{1+\frac{5}{6}}\right) - \frac{2}{5}

**Step 1: Simplify the expression inside the parentheses**

The term inside the parentheses is \frac{1}{1+\frac{5}{6}}. Simplify this by first finding the denominator:

1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6}

Thus, the expression inside the parentheses becomes:

\frac{1}{\frac{11}{6}} = \frac{6}{11}

**Step 2: Substitute the simplified expression back into the original**

Now, substitute \frac{6}{11} back into the main expression:

\frac{4}{5} + \frac{6}{11} - \frac{2}{5}

**Step 3: Find a common denominator**

The denominators of the fractions are 5, 11, and 5. The least common multiple of these numbers is 55.

**Step 4: Convert each fraction to have the denominator of 55**

• \frac{4}{5} = \frac{4 \times 11}{5 \times 11} = \frac{44}{55}
• \frac{6}{11} = \frac{6 \times 5}{11 \times 5} = \frac{30}{55}
• \frac{2}{5} = \frac{2 \times 11}{5 \times 11} = \frac{22}{55}

**Step 5: Perform the addition and subtraction**

Add and subtract these fractions:

\frac{44}{55} + \frac{30}{55} - \frac{22}{55} = \frac{44 + 30 - 22}{55} = \frac{52}{55}

Thus, the value of the expression is \frac{52}{55}, which matches the correct answer.