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

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Solution

Given $(1 + x) (1 + x^{2}) (1 + x^{3}) . . . . . (1 + x^{100})$ , we have to find coefficient of $x^{9}$
$x^{9}$ can be written as $x^{9}, x^{1 + 8}, x^{2 + 7}, x^{3 + 6}, x^{4 + 5}, x^{1 + 2 + 6}, x^{1 + 3 + 5}, x^{2 + 3 + 4}$
Now coefficient are 1 in each case, and there are 8 terms
$∴$ coefficient of $x^{9} = 1 \times 8 = 8$

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