The coefficient of $x^{101}$ in the expression ${(5 + x)}^{500} + x {(5 + x)}^{499} + x^{2} {(5 + x)}^{498} + .......... x^{500}, x > 0,$ is

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$^{501} C_{100} {(5)}^{400}$

$^{501} C_{101} {(5)}^{399}$

$^{501} C_{101} {(5)}^{401}$

$^{501} C_{101} {(5)}^{400}$

answer is A.

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Detailed Solution

(A) Given expression is ${(5 + x)}^{500} + x {(5 + x)}^{499} + x^{2} {(5 + x)}^{498} + ....... x^{500}$
Take ${(5 + x)}^{500}$ common, then it forms G.P.
$\begin{array}{l} [{(5 + x)}^{500} ({(\frac{x}{5 + x})}^{501} - 1)] (\frac{x}{5 + x}) - 1 \\ = \frac{{(5 + x)}^{501} - x^{501}}{5} \end{array}$
$\Rightarrow$ Coefficient $x^{101}$ in given expression is calculated by $(\frac{1}{5} {(x + 5)}^{501})$ only $= \frac{^{501} C_{101} 5^{400}}{5} =^{501} C_{101} 5^{399}$