The coefficient of xk in the expansion of 1-2x-x2e-x is

1-2x-x2e-x =(1-2x-x2) ex =(1-2x-x2) 1+x+x22+x33+....xk coef=11k-21k-1-1k-2 =1-2k-k(k-1)k =1-k-k2k

The coefficient of xk in the expansion of 1-2x-x2e-x is

1-2x-x2e-x =(1-2x-x2) ex =(1-2x-x2) 1+x+x22+x33+....xk coef=11k-21k-1-1k-2 =1-2k-k(k-1)k =1-k-k2k

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The coefficient of $x^{k}$ in the expansion of $\frac{1 - 2 x - x^{2}}{e^{- x}}$ is

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$\frac{1 - k - k^{2}}{k!}$

$\frac{1}{k!}$

$\frac{k^{2} + 1}{k!}$

$\frac{1 - k}{k!}$

answer is A.

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Detailed Solution

$\begin{array}{rcl} \frac{1 - 2 x - x^{2}}{e^{- x}} \\ = & (1 - 2 x - x^{2}) e^{x} \\ = & (1 - 2 x - x^{2}) (1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . . .) \\ x^{k} co e f & = & 1 (\frac{1}{k}) - 2 (\frac{1}{k - 1}) - (\frac{1}{k - 2}) \\ = & \frac{1 - 2 k - k (k - 1)}{k} \\ = & \frac{1 - k - k^{2}}{k} \end{array}$