The coefficient of $x^{9}$ in the expansion of $(1 + x) (1 + x^{2}) (1 + x^{3}) \dots \dots . (1 + x^{100})$ is .

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Solution

The number of possible combination for $x^{9}$ will be
$x^{0} \cdot x^{9} x^{1} \cdot x^{8} x^{2} \cdot x^{7} x^{3} \cdot x^{6} x^{4} \cdot x^{5} x^{1} \cdot x^{2} \cdot x^{6} x^{1} \cdot x^{3} \cdot x^{5} x^{2} \cdot x^{3} \cdot x^{4}$
Coefficient of all the terms will be $= 1$
Total cofficient $= 8$

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