The coefficient of $x^7$ in the expansion of $\left(1-x-x^2+x^3\right)^6$ is

The coefficient of $x^7$ in the expansion of $\left(1-x-x^2+x^3\right)^6$ is
  1. $-132$
  2. $-144$
  3. $132$
  4. $144$

Solution

$ \begin{aligned} & {\left[1-\mathrm{x}-\mathrm{x}^2(1-\mathrm{x})\right]^6=(1-\mathrm{x})^6\left(1-\mathrm{x}^2\right)^6} \\ & =\left[{ }^6 \mathrm{C}_0-{ }^6 \mathrm{C}_1 \mathrm{x}+{ }^6 \mathrm{C}_2 \mathrm{x}^2-{ }^6 \mathrm{C}_3 \mathrm{x}^3+{ }^6 \mathrm{C}_4 \mathrm{x}^4-{ }^6 \mathrm{C}_5 \mathrm{x}^5+{ }^6 \mathrm{C}_6 \mathrm{x}^6\right] \times\left[{ }^6 \mathrm{C}_0-{ }^6 \mathrm{C}_1 \mathrm{x}^2+{ }^6 \mathrm{C}_2 \mathrm{x}^4-{ }^6 \mathrm{C}_3 \mathrm{x}^6+\ldots .\right] \end{aligned} $ Coefficient of $\mathrm{x}^7={ }^6 \mathrm{C}_1{ }^6 \mathrm{C}_3-{ }^6 \mathrm{C}_3{ }^6 \mathrm{C}_2+{ }^6 \mathrm{C}_5{ }^6 \mathrm{C}_1=120-300+36=-144$

Asked in: JEE Main 2011

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