The coefficient of $x^{50}$ in the expansion of $(1 + x)^{1000} + 2 x (1 + x)^{999} + 3 x^{2} (1 + x)^{998} + . . . . . . + 1001 x^{1000}$

^{1000} C_{50}

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^{1001} C_{50}

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^{1002} C_{50}

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2^{1001}

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Solution

The correct option is C $^{1002} C_{50}$
Let, $S = (1 + x)^{1000} + 2 x (1 + x)^{999} + 3 x^{2} (1 + x)^{998} + . . . . . . + 1001 x^{1000}$
$\frac{x}{1 + x} S = x (1 + x)^{999} + 2 x^{2} (1 + x)^{998} + . . . . . . . . + 1000 x^{1000} + \frac{1001 x^{1001}}{1 + x}$
Substract above equations.
$(1 - \frac{x}{1 + x}) S = (1 + x)^{1000} + (1 + x)^{999} + x^{2} (1 + x)^{998} + . . . . + x^{1000} - \frac{1001 x^{1001}}{1 + x}$
$\Rightarrow S = (1 + x)^{1001} + x (1 + x)^{1000} + x^{2} (1 + x)^{999} + . . . . . . + x^{1000} (1 + x) - 1001 x^{1001}$
$= \frac{(1 + x)^{1001} [{(\frac{x}{1 + x})}^{1001} - 1]}{\frac{x}{1 + x} - 1} - 1001 x^{1001}$
[Sum of G.P]
$= (1 + x)^{1002} - x^{1002} - 1002 x^{1001}$
$∴$ coefficient of $x^{50}$ in $S =$ coefficient of $x^{50}$ in $[(1 + x)^{1002} - x^{1002} - 1002 x^{1001}] =^{1002} C_{50}$

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