$\text{The value of}\mathrm{ 10}+3\frac{3}{5}÷[2\frac{2}{3}÷\{7\frac{1}{6}-(2\frac{1}{2}+\frac{2}{3})\left\}\right]\text{is equal to:}$

The correct option is $\frac{77}{5}$.

To evaluate the given mathematical expression, follow the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction):

$10+3\frac{3}{5}÷[2\frac{2}{3}÷\{7\frac{1}{6}-(2\frac{1}{2}+\frac{2}{3})\left\}\right]$

Step 1: Convert all mixed fractions into improper fractions:

$3\frac{3}{5}=\frac{18}{5}$

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

$7\frac{1}{6}=\frac{43}{6}$

$2\frac{1}{2}=\frac{5}{2}$

Step 2: Solve the expression inside the innermost round parentheses $\left(\right)$:

$\frac{5}{2}+\frac{2}{3}=\frac{15+4}{6}=\frac{19}{6}$

Step 3: Solve the expression inside the curly braces $\left\{\right\}$:

$\frac{43}{6}-\frac{19}{6}=\frac{43-19}{6}=\frac{24}{6}=4$

Step 4: Solve the expression inside the square brackets $\left[\right]$:

$\frac{8}{3}÷4=\frac{8}{3}\times \frac{1}{4}=\frac{2}{3}$

Step 5: Perform the division:

$\frac{18}{5}÷\frac{2}{3}=\frac{18}{5}\times \frac{3}{2}=\frac{27}{5}$

Step 6: Perform the final addition:

$10+\frac{27}{5}=\frac{50+27}{5}=\frac{77}{5}$

Thus, the required value is $\frac{77}{5}$.