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If (1 + x)^(n) = C(0) + C(1) x + C(2) x^...

If `(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) +…+ C_(n) x^(n)` , find the values of the following . `sum_(i=0)^(n) sum_(j=0)^(n) (i+j) C_(i) C_(j)`

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`sum_(i=0)^(n) sum_(j=0)^(n) (i + j) C_(i) C_(j) = sum_(i=0)^(n) sum_(j=0)^(n) iC_(i) C_(j) + sum_(i=0)^(n) sum_(j=0)^(n) jC_(i) C_(j)`
`=sum_(i=0)^(n)i C_(i) (sum_(j=0)^(n) C_(j)) + sum_(j=0)^(n) jC_(j) (sum_(i=0)^(n)C_(i))`
`= sum_(i=0)^(n)i C_(i)(2^(n)) + sum_(i=0)^(n)j C_(j)(2^(n))`
` = 2^(n) sum_(i=0)^(n)i ""^(n)C_(i) + 2^(n) sum_(j=0)^(n) j""^(n)C_(j) `
`=2^(n) sum_(i=0)^(n)i.(n)/(i) .""^(n-1)C_(i-1)+ 2^(n) sum_(j=0)^(n)j * (n)/(j) * ""^(n-1)C_(j-1)`
`=n.2^(n) sum_(i=0)^(n) .""^(n-1)C_(i-1)+ n.2^(n) sum_(j=0)^(n) * ""^(n-1)C_(j-1)`
`= n.2^(n).2^(n-1) + n.2^(n) * 2^(n-1)`
` = n*2*2^(2n-1) = n*2^(2n)` .
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