Q59 / 160 · AP EAMCET 2019 (23 Apr Shift 1) · Mathematics

The coefficient of \(x^4\) in the expansion of \(\frac{1}{(1-x)(1-2 x)(1-3 x)}\) is

  • A
    602
  • Correct answer
    B
    301
  • C
    \(\frac{601}{2}\)
  • D
    302

Solution

\(\begin{aligned} & \text {Since, } \frac{1}{(1-x)(1-2 x)(1-3 x)} \\ & =(1-x)^{-1}(1-2 x)^{-1}(1-3 x)^{-1} \\ & =\left(1+x+x^2+x^3+x^4\right)\left(1+2 x+4 x^2+8 x^3+16 x^4\right) \\ & \left(1+3 x+9 x^2+27 x^3+81 x^4\right) \end{aligned}\) [Expand till \(x^4\)-term because coefficient of \(x^4\) is required and \((1-a x)^{-n}\) \(\left.=1+a x+a^2 x^2+a^3 x^3+\ldots \ldots . . . . . .\right]\) So, coefficient of \(x^4\) is \(\begin{aligned} & 81+54+36+24+16+27+18+12+8+9+ \\ & 6+4+3+2+1=301 \end{aligned}\) Hence, option (b) is correct.

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

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