The coefficient of x3 in the expansion of (1−x+x2)6 is

(1−x+x2)6=[1−x(1−x)]6 =6C0−6C1x(1−x)+6C2x2(1−x)2−6C3x3(1−x)3+…to 7 terms =6C0−6C1x(1−x)+6C2x2(1−2x+x2)−6C3x3(1−3x+3x2−x3)+…to 7 terms∴Coefficient of x3=−2⋅6C2−6C3 (collecting coefficients of x3 from each term)=−26!2!4!−6!3!3!=−50

The coefficient of x3 in the expansion of (1−x+x2)6 is

(1−x+x2)6=[1−x(1−x)]6 =6C0−6C1x(1−x)+6C2x2(1−x)2−6C3x3(1−x)3+…to 7 terms =6C0−6C1x(1−x)+6C2x2(1−2x+x2)−6C3x3(1−3x+3x2−x3)+…to 7 terms∴Coefficient of x3=−2⋅6C2−6C3 (collecting coefficients of x3 from each term)=−26!2!4!−6!3!3!=−50

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The coefficient of x³ in the expansion of ${(1 - x + x^{2})}^{6}$ is

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Detailed Solution

$\begin{aligned} {(1 - x + x^{2})}^{6} = [1 - x (1 - x)]^{6} \\ =^{6} C_{0} -^{6} C_{1} x (1 - x) +^{6} C_{2} x^{2} (1 - x)^{2} -^{6} C_{3} x^{3} (1 - x)^{3} + \dots to 7 terms \\ =^{6} C_{0} -^{6} C_{1} x (1 - x) +^{6} C_{2} x^{2} (1 - 2 x + x^{2}) -^{6} C_{3} x^{3} (1 - 3 x + 3 x^{2} - x^{3}) + \dots to 7 terms \end{aligned}$