The coefficient of x7 in the expansion of 1−x−x2+x36 is

A
132
B
144
C
-132
D
-144

Solution:

${(1 - x - x^{2} + x^{3})}^{6} = (1 - x)^{6} {(1 - x^{2})}^{6}$

coefficient of $x^{7}$ in $(1 - x)^{6} {(1 - x^{2})}^{6}$

= coefficient of $x^{7}$ in $[1 -^{6} C_{1} x +^{6} C_{2} x^{2} - \dots +^{6} C_{6} x^{6}]$

$\begin{array}{l} \times [1 -^{6} C_{1} x^{2} +^{6} C_{2} x^{4} - \dots +^{6} C_{6} x^{12}] \\ = (-^{6} C_{1}) (-^{6} C_{3}) + (-^{6} C_{3}) (^{6} C_{2}) + (-^{6} C_{5}) (-^{6} C_{1}) \\ = - 144 . \end{array}$

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

Solution:

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