In the expansion of ${(1 - x - x^{2} + x^{3})}^{6},$ the coefficient of $x^{7}$ is

- 144

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120

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- 128

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- 120

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Solution

The correct option is A $- 144$
${(1 - x - x^{2} + x^{3})}^{6}$
$= (1 - x)^{6} {(1 - x^{2})}^{6}$
$= (6 \sum r = 0^{6} C_{r} (- 1)^{6 - r} x^{6 - r}) (6 \sum s = 0^{6} C_{s} (- 1)^{6 - s} x^{12 - 2 s})$
For $x^{7}$ term, the possible combinations are
$r = 1, s = 5$
$r = 3, s = 4$
$r = 5, s = 3$
Hence, coefficient of $x^{7}$ is
$^{6} C_{1} (- 1)^{6 - 1} \cdot^{6} C_{5} (- 1)^{6 - 5} +^{6} C_{3} (- 1)^{6 - 3} \cdot^{6} C_{4} (- 1)^{6 - 4} +^{6} C_{5} (- 1)^{6 - 5} \cdot^{6} C_{3} (- 1)^{6 - 3}$
$= (- 6) (- 6) + (- 20) (15) + (- 6) (- 20)$
$= - 144$

Alternatively, $(1 - x)^{6} {(1 - x^{2})}^{6}$
$= (1 - 6 x + 15 x^{2} - 20 x^{3} + 15 x^{4} - 6 x^{5} + x^{6}) (1 - 6 x^{2} + 15 x^{4} - 20 x^{6} + 15 x^{8} - 6 x^{10} + x^{12})$
$∴$ Coefficient of $x^{7}$ is $(- 6) (- 20) + (- 20) (15) + (- 6) (- 6)$
$= - 144$

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