Q57 / 160 · AP EAMCET 2018 (24 Apr Shift 1) · Mathematics

The coefficient of $x^4$ in the expansion of $\left(1+x-x^2-x^3\right)^{11}$ is

  • A
    990
  • B
    220
  • Correct answer
    C
    − 220
  • D
    − 385

Solution

$$ \begin{array}{ll} & \left(1+x-x^2-x^3\right)^{11} \\ \Rightarrow & {\left[1(1+x)-x^2(1+x)\right]^{11}} \\ \Rightarrow & {\left[\left(1-x^2\right)(1+x)\right]^{11}} \\ \Rightarrow & {\left[(1-x)(1+x)^2\right]^{11}} \\ \Rightarrow & (1-x)^{11} \cdot(1+x)^{22} \end{array} $$ The coefficient of $x^4$ can be given, $$ \begin{aligned} & =\sum_{r=0}^4(-1)^r \cdot{ }^{11} C_r \cdot{ }^{22} C_{4-r} \\ & \Rightarrow{ }^{11} C_0 \times{ }^{22} C_4-{ }^{11} C_1 \cdot{ }^{22} C_3 \\ & +{ }^{11} C_2 \cdot{ }^{22} C_2-{ }^{11} C_3 \cdot{ }^{22} C_1+{ }^{11} C_4 \cdot{ }^{22} C_0 \\ & \Rightarrow 7315-16940+12705-3630+330 \\ & \Rightarrow \quad 20350-20570=-220 \\ & \end{aligned} $$

Hence, the correct answer is option: 'C'.

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