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Find the sum of the series .^(84)C(4)+6x...

Find the sum of the series `.^(84)C_(4)+6xx.^(84)C_(5)+15xx.^(84)C_(6)+20xx.^(84)C_(7)+15xx.^(84)C_(8)+6xx.^(84)C_(9)+.^(84)C_(10)`.

Text Solution

Verified by Experts

The correct Answer is:
`.^(90)C_(10)`
`.^(84)C_(4) + 6 xx .^(84)C_(5) + 15 xx .^(84)C_(6)+20 xx .^(84)C_(7)+15 xx.^(84)C_(8)+6xx.^(84)C_(9)+.^(84)C_(10)`
`= .^(6)C_(6) xx .^(84)C_(4)+ .^(6)C_(5) xx .^(84)C_(5) + .^(6)C_(4) xx .^(84)C_(6)+ .^(6)C_(3) xx .^(84)C_(7) + .^(6)C_(2) xx .^(84)C_(8)+ .^(6)C_(1)xx.^(84)C_(9) + .^(6)C_(0) xx.^(84)C_(10)`
`=` Coefficient of `x^(10)` in `(1+x)^(6)(1+x)^(84)`
`=` Coefficient of `x^(10)` in `(1+x)^(90)`
`= .^(90)C_(10)`
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