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 .
Coefficient of ` x^(7)` in the expansion of
` (1 + x)^(2) (3 + x)^(3) (5 + x)^(4)` is

Options

A. 112
B. 224
C. 342
D. 416

Answer

d

Explanation

` (x + 1)^(2) (x + 3)^(3) (x + 5)^(4) `
` = (x + 1) (x + 1) (x + 3) (x + 3) ( x+ 3) (x + 5) (x + 5) (x + 5) (x + 5) `
` = x^(9) + (1 + 1 + 1 + 3 + 3 + 3 + 5 + 5 + 5 ) x^(8) `
`+ (1*1+ 1*3+ 1*3+1 *5+ .. . + 1*5 + 3*3`
` + 3*3 + 3*5 + ...+ 5+5)x^(7) + ... `
` therefore ` Coefficient of ` x^(7) = (1 + 1 + 3 + 3 + 3 + 5 + 5 + 5 +5 )^(2)`
`(-(1^(2) + 1^(2) +3^(2)+ 3^(2) + 5^(2) + 5^(2) + 5^(2) + 5^(2)))/(2) `
` = ((31)^(2) - (129))/(2) = (961- 129)/(2) = 416`