The co-efficient of $x^{9}$ in the expansion of $(1 + x) (1 + x^{2}) (1 + x^{3}) . . . (1 + x^{100})$ is

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Solution

$9 = (0, 9), (1, 8), (2, 7), (3, 6), (4, 5) =$ 5 ways
$9 = (1, 2, 6), (1, 3, 5), (2, 3, 4) =$ 3 ways
Thus coeff. of $x^{9}$ in the given expression is $8$ .

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