Question

The coefficient of $x^n$ in the polynomial $(x+{}^n{C}_0)(x+3\cdot {}^n{C}_1)\cdots (x+(2n+1)\cdot {}^n{C}_n)$ is

• A. $n\cdot 2^n$
• B. $n\cdot 2^{n+1}$
• C. $(n+1)2^n$
• D. $n\cdot 2^n+1$

Answer 1

The correct option is C $(n+1)2^n$

Coefficient of $x^n$ is
$[{}^n{C}_0+3\cdot {}^n{C}_1+5\cdot {}^n{C}_2+\dots +(2n+1)\cdot {}^n{C}_n]=\sum _{r=0}^n(2r+1){}^n{C}_r=2\sum _{r=0}^{n}r{}^n{C}_r+\sum _{r=0}^n{}^n{C}_r=2\sum _{r=0}^{n}r{}^n{C}_r+2^n$

We know that,
$(1+x{)}^n=1+{}^n{C}_{1}x+{}^n{C}_2{x}^2+\dots +{}^n{C}_n{x}^n$
Differentiating w.r.t. $x$, we get
$\Rightarrow n(1+x{)}^{n-1}={}^n{C}_1+2{}^n{C}_{2}x+\dots +n{}^n{C}_n{x}^{n-1}$
Putting $x=1$
$\Rightarrow n\left(2^{n-1}\right)={}^n{C}_1+2{}^n{C}_2+\dots +n{}^n{C}_n\Rightarrow n\left(2^{n-1}\right)=0\left({}^n{C}_0\right)+{}^n{C}_1+2{}^n{C}_2+\dots +n{}^n{C}_n\Rightarrow n\left(2^n\right)=2\sum _{r=0}^{n}r{}^n{C}_r$

So, the required coefficient
$=n\left(2^n\right)+2^n=2^n(n+1)$