In the expansion of `(x+x^(2)+x^(3)+…….) (1+x+x^(2)+x^(3)) (x^(2)+x^(3)+x^(4)+……+x^(10))` the coefficient of `x^(6)` is

Options

A. `10`
B. `18`
C. `30`
D. `12`

Answer

A

Explanation

In the expansion ………..
`(x+x^(2)+…….)………`
Coefficient of `x^(6)` in the expansion `(x+x^(2)+…..)`
`(1+x+x^(2)+x^(3))(x^(2)+…….+x^(10))`
= Coefficient of `x^(6)` in the expension
`x^(3)(1+x+x^(2)+……) (1+x+x^(2)+x^(3))(1+x+x^(2)+....+x^(8))`
= Coefficient of `x^(3)` in the expansion `(1-x^(4))(1-x^(9))(1-x)^(-3)`
= Coefficient of `x^(3)` in the expansion `(1-x)^(-3) = ^(5)C_(3) = 10`