Question

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

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The correct option is C

8

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