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Evaluate: (i) .^(12)C(1)+.^(12)C(2)+.^...

Evaluate:
(i) `.^(12)C_(1)+.^(12)C_(2)+.^(12)C_(3)+ . . .+.^(12)C_(12)`
(ii) `.^(19)C_(3)+.^(19)C_(5)+.^(19)C_(7)+ . . .+^(19)C_(19)`.

Text Solution

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(i) `.^(12)C_(1)+.^(12)C_(2)+.^(12)C_(3)+ . . .+.^(12)C_(12)`
`=.^(12)C_(0)+.^(12)C_(1)+.^(12)C_(2)+.^(12)C_(3)+ . . . .+.^(12)C_(12)-.^(12)C_(0))`
={sum of coefficients in the expansion of `(1+a)^(12)`}`-.^(12)C_(1)`
={sum of coefficient of evenn terms in the expansion of `(1+a)^(19)`}-`.^(19)C_(1)=2^(19-1)-19`
`=2^(18)-19`
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