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

-144

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132

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144

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-132

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Solution

The correct option is A
-144

$(1 - x - x^{2} + x^{3})^{6} = (1 - x)^{6} (1 - x^{2})^{6} = (1 -^{6} C_{1} x +^{6} C_{2} x^{2} -^{6} C_{3} x^{3} +^{6} C_{4} x^{4} -^{6} C_{5} x^{5} +^{6} C_{6} x^{6}) \times (1 -^{6} C_{1} x^{2} +^{6} C_{2} x^{4} -^{6} C_{3} x^{6} +^{6} C_{4} x^{8} -^{6} C_{5} x^{10} +^{6} C_{6} x^{12})$
Coefficient of $x^{7} = (1 -^{6} C_{1} x -^{6} C_{3}) + (1 -^{6} C_{3} x -^{6} C_{2}) + (1 -^{6} C_{5} x -^{6} C_{1})$
=120-300+36=-144

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