To solve the expression \( \frac{11}{15} : \left( \frac{1}{10} - \frac{1}{2} \right) 211/15 : \left( \frac{1}{10} - \frac{1}{2} \right) 2 \), let's break it down step by step.

1. **Calculate \( \frac{1}{10} - \frac{1}{2} \)**:

To subtract these fractions, we first need a common denominator. The least common multiple of 10 and 2 is 10.

\[
\frac{1}{10} - \frac{1}{2} = \frac{1}{10} - \frac{5}{10} = \frac{1 - 5}{10} = \frac{-4}{10} = \frac{-2}{5}
\]

2. **Replace \( \frac{1}{10} - \frac{1}{2} \) in the expression**:

The expression now becomes:

\[
\frac{11}{15} : \left( \frac{-2}{5} \right) \cdot \frac{211}{15} : \left( \frac{-2}{5} \right) \cdot 2
\]

3. **Calculate \( \left( \frac{-2}{5} \right) \cdot \frac{211}{15} \)**:

\[
\left( \frac{-2}{5} \right) \cdot \frac{211}{15} = \frac{-2 \cdot 211}{5 \cdot 15} = \frac{-422}{75}
\]

4. **Calculate \( \left( \frac{-2}{5} \right) \cdot 2 \)**:

\[
\left( \frac{-2}{5} \right) \cdot 2 = \frac{-4}{5}
\]

5. **Now the expression is**:

\[
\frac{11}{15} : \left( \frac{-422}{75} \right) : \left( \frac{-4}{5} \right)
\]

The division by a fraction is equivalent to multiplying by its reciprocal.

6. **Convert the division to multiplication**:

\[
\frac{11}{15} \cdot \left( \frac{-75}{422} \right) \cdot \frac{5}{-4}
\]

Multiply this out:

\[
= \frac{11 \cdot 75 \cdot 5}{15 \cdot 422 \cdot 4}
\]

Simplifying:

- \( \frac{75}{15} = 5 \)
- Hence, \( 11 \cdot 5 \cdot 5 = 275 \)

The denominator becomes:

\[
422 \cdot 4 = 1688
\]

7. **So we now have**:

\[
\frac{275}{1688}
\]

8. **Final step: Simplify \( \frac{275}{1688} \)**:

The GCD of 275 and 1688 is 11.

- \( 275 \div 11 = 25 \)
- \( 1688 \div 11 = 153 \)

Thus:

\[
\frac{275}{1688} = \frac{25}{153}
\]

Therefore, the final result of the expression is:

\[
\frac{25}{153}
\]