Find the coefficient of `x^7` in the expansion of `(1 - x -x^2 + x^3)^(6)dot`

Options

A. 144
B. -132
C. -144
D. 132

Answer

c

Explanation

We have
`(1 - x - x^(2) + x^(3))^(6) = {(1 - x)( 1 - x^(2))}^(7) = (1 - x )^(7) = (1 - x)^(7) (1 - x^(2))^(7)`
= `(""^(6)C_(0) - ""^(6)C_(1) x + ""^(6)C_(2)x^(2) - ""^(6)C_(3)x^(3) +... + ""^(6)C_(6)x^(6))`
`(""^(6)C_(0) - ""^(6)C_(1) x^(2) + ""^(6)C_(2)x^(4) - ""^(6)C_(3)x^(6) +... )`
`therefore` Coefficient of `x^(7)` in `(1 - x - x^(2) + x^(3))^(6)`
` = - ""^(6)C_(1)xx (-""^(6)C_(3)) - ""^(6)C_(3)xx ""^(6)C_(3)xx ""^(6)C_(2) - ""^(6)C_(5) xx - (""^(6)C_(1))`
` 120 - 300 + 36 = -144` .