The co-efficient of $x^{6}$ in the series of $e^{2 x}$ is

The co-efficient of $x^{6}$ in the series of $e^{2 x}$ is
  1. $\frac{2}{45}$
  2. $\frac{7}{45}$
  3. $\frac{4}{45}$
  4. $\frac{1}{45}$

Solution

$e^{2 x}=1+2 x+\frac{(2 x)^{2}}{2 !}+\frac{(2 x)^{3}}{3 !}+\frac{(2 x)^{4}}{4 !}+\frac{(2 x)^{6}}{6 !}+$ $x^{6} \rightarrow$ coefficient $=\frac{2^{6} \cdot x^{6}}{6 !}=\frac{4}{45}$

Asked in: MHT CET 2020 (20 Oct Shift 2)

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