The coefficient of $x^{99}$ in the expansion of $(x - 1) (x - 2) (x - 3) . . . . . . . . (x - 100)$ is

5050

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5000

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-5050

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-5000

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Solution

The correct option is D -5050
We have, $(x - 1) (x - 2) (x - 3) . . . . . . . . . . . . (x - 100)$
$= x^{100} - (1 + 2 + 3 + . . . . . . . + 100) x^{99} + . . . . . . . . .$
$∴$ coefficient of $x^{99} = - \frac{100}{2} [2 + 99] = - 50 \times 101 = - 5050$
Hence, C is the correct option.

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