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If (.^(20)C(1))/(1)+(.^(20)C(3))/(2)+(.^...

If `(.^(20)C_(1))/(1)+(.^(20)C_(3))/(2)+(.^(20)C_(5))/(3)+.......+("^(20)C_(19))/(10)=(K(2^(20-1)))/(21)`, then K is equal to :

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The correct Answer is:
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`(.^(20)C_(1))/(1)+(.^(20)C_(3))/(2)+(.^(20)C_(5))/(3)+......+(.^(20)C_(19))/(10)`
`=underset(r=1)overset(10)Sigma(1)/(r).^(20)C_(2r-1)=(2)/(21)underset(r=1)overset(10)Sigma(21)/(2r).^(20)C_(2r-1)`
`=(2)/(21)underset(r=1)overset(10)Sigma.^(21)C_(2r)=(2)/(21)[.^(21)C_(2)+.^(21)C_(4)+.^(21)C_(20)]`
`=(2)/(21)(2^(20)-1)`
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