Find the term independent of `x` in the expansion of `(1+x+2x^3)[(3x^2//2)-(1//3)]^9`

Options

A. `1//3`
B. `19//54`
C. `17//54`
D. `1//4`

Answer

c

Explanation

We have,
`(1 + x + 2x^(3))((3)/(2) x^(2)-(1)/(3x))^(9)`
`(1 + x + 2x^(3)){sum_(r=0)^(9)""^(9)C_(r) ((3)/(2) x^(2))^(9-r)(-(1)/(3x))^(r)}`
`(1 + x + 2x^(3)){sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(18 - 3r)}`
`= {sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(18 - 3r)}`
`+ {sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(19 - 3r)}`
`+2 {sum_(r=0)^(9)""^(9)C_(r) ((3)/(2))^(9-r)(-(1)/(3))^(r)x^(21 - 3r)}`
Clearly, first and third expansions contain term idepen-
dent of x and obtained by equation 18 - 3r = 0 and 21 -3r = 0
respectivley.
So, coefficient of the term independent of x
`""^(9)C_(6) ((3)/(2))^(9-6)(1-(1)/(3))^(6) +2{""^(9)C_(7)((3)/(2))^(9-7) (-(1)/(3))^(7)}`
` (7)/(18) - (2)/(27) = (17)/(54)` .