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Evaluate the following : (i) 1+.^(20)C...

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)`

Text Solution

AI Generated Solution

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 ...
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