Question

The coefficient of $x^{n-1}$ in the polynomial $\left(x{+}^{2n+1}{C}_0\right)\left(x{+}^{2n+1}{C}_1\right)\left(x{+}^{2n+1}{C}_2\right)\cdots \left(x{+}^{2n+1}{C}_n\right)$ is

• A. $2^{2n+1}-1$
• B. $2^{2n}$
• C. $2^{2n+1}$
• D. None of these

Answer 1

Answer: (B)

$2^{2n}$

$(x+{}^{2n+1}{C}_0)(x+{}^{2n+1}{C}_1)(x+{}^{2n+1}{C}_2)+...(x+{}^{2n+1}{C}_n)$
$=x^n+[{}^{2n+1}{C}_0+{}^{2n+1}{C}_1+...{}^{2n+1}{C}_n]x^{n-1}+...$
Hence
Coefficient of $x^{n-1}$
$={}^{2n+1}{C}_0+{}^{2n+1}{C}_1+...{}^{2n+1}{C}_n$
$=\frac{1}{2}\left[2\right({}^{2n+1}{C}_0+{}^{2n+1}{C}_1+...{}^{2n+1}{C}_n\left)\right]$
$=\frac{1}{2}[{}^{2n+1}{C}_0+{}^{2n+1}{C}_1+...{}^{2n+1}{C}_n+{}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_{n+2}+...{}^{2n+1}{C}_{2n+1}]$
Since ${}^n{C}_r={\frac{n}{C}}_{n-r}$
Now
$\frac{1}{2}[{}^{2n+1}{C}_0+{}^{2n+1}{C}_1+...{}^{2n+1}{C}_n+{}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_{n+2}+...{}^{2n+1}{C}_{2n+1}]$
$=\frac{1}{2}\left[\right(1+x{)}^{2n+1}{]}_{x=1}$
$=\frac{1}{2}\left[2^{2n+1}\right]$
$=2^{2n}$.