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  4. The coefficient of x7 in the expansion of (1- x - x2 + x3 )6 is

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Q. The coefficient of $x^7$ in the expansion of $(1- x - x^2 + x^3 )^6$ is

AIEEEAIEEE 2011Binomial Theorem

Solution:

$(1 - x - x^2 + x^3)^6 = [(1- x) - x^2 (1 - x)]^6$
$ = (1- x)^6 (1 - x^2)^6$
$= (1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6)$
$ \times (1 - 6x^2 + 15x^4 - 20x^6 + 15x^8 - 6x^{10} + x^{12})$
Coefficient of $x^7 = (- 6) (- 20) + (- 20)(15) + (- 6) (-6) = - 144$