Evaluate the following :
`(i) 1+.^(20)C_(1)+^(20)C_(2)+^(20)C_(3)+....+^(20)C_(19)+^(20)C_(20)`
`(ii) ^(10)C_(1)+^(10)C_(2)+^(10)C_(3)+.....+^(10)C_(9)`
`(iii)^(25)C_(1)+^(25)C_(3)+^(25)C_(5)+.....+^(25)C_(25)`
`(iv) ^(18)C_(2)+^(18)C_(4)+^(18)C_(4)+^(18)C_(6)+....+^(18)C_(18)`

Explanation

To evaluate the given expressions, we will use some properties of binomial coefficients and the binomial theorem.

### (i) Evaluate \( 1 + \binom{20}{1} + \binom{20}{2} + \binom{20}{3} + \ldots + \binom{20}{20} \)

Using the property of binomial coefficients, we know that:

\[
\sum_{k=0}^{n} \binom{n}{k} = 2^n
...