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) = underset(1lei lt j le n)(sumsum) a_(i) a_(j) , S_(3) underset(1le i ltk le n)(sumsumsum)a_(i) a_(j) a_(k)`
and so on .
Coefficient of ` x^(7)` in the expansion of
` (1 + x)^(2) (3 + x)^(3) (5 + x)^(4)` is

Options

A. ` n *2^(n)`
B. `(n +1)*2^(n)`
C. `n *2^(n+1)`
D. ` n *2^(n) +1`

Answer

b

Explanation

`(x + C_(0)) (x + 3 .C_(1)) (x + 5. C_(2)) + …{ x + (2n + 1). C_(n)}`
` = x^(n+1) + x^(n) { C_(0) + 3 . C_(1) + 5 . C_(2) + …+ (2n + 1). C_(n)}`
` therefore ` Coefficient of ` x^(n) = C_(0) + 3 . C_(1) + 5 . C_(2) + … + (2n +1) . C_(n) `
` = (C_(0) + C_(1) + C_(2) + ....+ C_(n)) + 2 { C_(1) + 2 . C_(2) + ....n . C_(n)} `
` = 2^(n) + 2 { n + 2 . (n(n-1))/(2) + ...+ n} `
` = 2^(n) + 2n { 1 + (n-1) + ((n-1)(n-2))/(1.2)+ ...+ } `
` = 2^(n) + 2n { ""^(n-1)C_(0) + ""^(n-1)C_(1) + ""^(n-1)C_(2) + ....+ ""^(n+1)C_(n-1)} `
` = 2^(n) + 2n (1 + 1)^(n-1) = 2^(n)+ n. 2^(n) = (n +1)2^(n) `