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

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Solution

The correct option is C
8

Coefficient of $x^{9}$ in the expansion of
$1 + x) ((1 + x^{2})$ $(1 + x^{3})$ . . . $(1 + x^{100})$ = Terms having $x^{9}$
$= [1^{99} . x^{9}, 1^{98} . x . x^{8}, 1^{98} . x^{2} . x^{7}, 1^{98} . x^{3} . x^{6}, 1^{98} . x^{4} . x^{5}, 1^{97} . x . x^{2} . x^{6}, 1^{97} . x . x^{3} . x^{5}, 1^{97} . x^{2} . x^{3} . x^{4}]$
$∴$ Coefficient of $x^{9}$ = 8

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