Question

The cofficient of $x^9$ in the polynomial given by $\sum _{r=1}^{11}(x+r)(x+r+1)(x+r+2).......(x+r+9)$ is

• A. $5511$
• B. $5151$
• C. $1515$
• D. $1155$

Answer 1

The correct option is D $1155$

Here, there are $10$ different brackets in which power of $x$ is $1$. We need to find the coefficient of $x^9$. To find that we will select $x$ from any $9$ brackets and select the constant term from the remaining bracket.
Ex: If the constant term is selected from the first bracket the term is $r\cdot x^9$, if it is from second bracket the term is $(r+1)\cdot x^9$ and so on

So, the coefficient of $x^9$ is
$=\sum _{r=1}^{11}r+(r+1)+(r+2)+....+(r+9)$
$=\sum _{r=1}^{11}10r+45=1155$