The coefficient of $x$ in the expansion of $(1 + x) (1 + 2 x) (1 + 3 x) . . . . . . (1 + 100 x)$ is

$5050$

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$10100$

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$5151$

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$4950$

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Solution

The correct option is A
$5050$

Explanation for the Correct Answer:
Find the coefficient of $x$ :
Given expression is $(1 + x) (1 + 2 x) (1 + 3 x) . . . . . . (1 + 100 x)$
On multiplying the first two terms we see that $(1 + x) (1 + 2 x) = (1 + (1 + 2) x + 2 x^{2})$ , so the coefficient of $x$ is $1 + 2$
On multiplying the first three terms we see that $(1 + x) (1 + 2 x) (1 + 3 x) = (1 + (1 + 2 + 3) x + 11 x^{2} + 6 x^{3})$ ,
so the coefficient of $x$ is $1 + 2 + 3$
Hence, for $100$ terms, we can see that the Coefficient of $x$ is
$= 1 + 2 + 3 + . . . . . + 100 = \frac{100 \times 101}{2} = 5050$
Hence, option (A) is the correct answer.

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