The value of $p$ , for which coefficient of $x^{50}$ in the expression $(1 + x)^{1000} + 2 x (1 + x)^{999} + 3 x^{2} (1 + x)^{998} + . . . . + 1001 x^{1000}$ is equal to $^{1002} C_{p}$ , is

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Solution

Let $S = \sum_{r = 0}^{1000} (1 + r) x^{r} (1 + x)^{1000 - r}$

So, $S - \frac{x}{(1 + x)} S = (\sum_{r = 0}^{1000} x^{r} (1 + x)^{1000 - r}) - 1001 \frac{x^{1001}}{(1 + x)} = (\frac{(1 + x)^{1000} (1 - (\frac{x}{1 + x})^{1001})}{1 - \frac{x}{1 + x}}) - 1001 \frac{x^{1001}}{1 + x}$

$\Rightarrow S (1 - \frac{x}{1 + x}) = ((1 + x)^{1001} - x^{1001}) - 1001 \frac{x^{1001}}{1 + x}$
$\Rightarrow S = (1 + x) ((1 + x)^{1001} - x^{1001}) - 1001 x^{1001} = (1 + x)^{1002} - 1002 x^{1001} - x^{1002}$

So, $S = (\sum_{r = 0}^{1002}^{1002} C_{r} x^{r}) - 1002 x^{1001} - x^{1002}$

So, coefficient of $x^{50}$ in the expansion will be, $^{1002} C_{50}$

So, $p = 50$ or $p = 1002 - 50 = 952$ ( $∵^{n} C_{r} =^{n} C_{n - r}$ ).

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