1.

If(x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where ,S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n) a_(i) a_(j) a_(k) and so on . If (1 + x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n) x^(n) the cefficient ofx^(n) in the expansion of (x + C_(0))(x + C_(1)) (x + C_(2))...(x + C_(n))is

Answer»

`2^(2n-1) - (1)/(2) ""^(2n)C_(n-1)`
`2^(2n-1) - (1)/(2) ""^(2n)C_(n)`
`2^(2n-1) - (1)/(2) ""^(2n+1)C_(n)`
`2^(2n-1) - (1)/(2) ""^(2n+1)C_(n-1)`

SOLUTION :`(x + C_(0)) (x + C_(1)) (x + X_(2)) + …+ (x + C_(n)) `
` = x^(n+1) + (sum_(r=0)^(n) C_(r))^(n) + ( UNDERSET(0 le i j le n )(sumsum)C_(i) C_(j)) x^(n-1) + ... `
` therefore ` COEFFICIENTOF ` x^(n-1)" in" underset(0 le i j le n )(sumsum)C_(i) C_(j)`
`= (1)/(2){ (sum_(r=0)^(n) C_(r))^(2) - ( sum_(r=0)^(n) C_(r)^(2) )} = (1)/(2) = { 2^(2n) - ""^(2n)C_(n)} `
` = 2^(2n-1)- (1)/(2) . ""^(2n)C_(n) ` .


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