Question

Coefficient of $x^{65}$ in expansion of ${\prod }_{i=0}^{10}{(i+x)}^{i+1}$ is-

• A. $650$
• B. $325$
• C. $440$
• D. $66$

Answer 1

The correct option is C $440$

${\prod }_{i=0}^{10}{(i+x)}^{i+1}=x{(1+x)}^2{(2+x)}^3......{(10+x)}^{11}$
$=x^{66}+(1.2+2.3+3.4+.......+10.11)x^{65}+........$
$\therefore$ Coefficient of $x^{65}$ is $={\sum }_{n=1}^{10}n(n+1)={\sum }_{n=1}^{10}{n}^2+{\sum }_{n=1}^{10}n=385+55=440$