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

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Question

The coefficient of

x^{7}

in the expansion of

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

is?

Options:

A . 132

B . 144

C . -132

D . -144

Answer: Option D
:
D

(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}) (1 -^{6} C_{1} x^{2} +^{6} C_{2} x^{4} -^{6} C_{3} x^{6} + \dots)

\Rightarrow

coefficient of

x^{7}

is

^{6} C_{1} .^{6} C_{3} -^{6} C_{3} .^{6} C_{2} +^{6} C_{5} .^{6} C_{1} = 6 \times 20 - 20 \times 15 + 6 \times 6 = - 144

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