The remainder when `(sum_(k=1)^(5) ""^(20)C_(2k-1))^(6)` is divided by 11, is :

Answer

To solve the problem of finding the remainder when \((\sum_{k=1}^{5} \binom{20}{2k-1})^6\) is divided by 11, we will follow these steps:

### Step 1: Evaluate the Summation
We need to evaluate the summation \(\sum_{k=1}^{5} \binom{20}{2k-1}\). This can be rewritten as:
\[ \binom{20}{1} + \binom{20}{3} + \binom{20}{5} + \binom{20}{7} + \binom{20}{9} \]