Question

The coefficient of $x^n$ in the polynomial
$\left(x+^{n}C_{0}\right)\left(x+3. ^{n}C_{1}\right)\left(x+5. ^{n}C_{2}\right) .... \left(x+\left(2n+1\right)^{n}C_{n}\right)$ is

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

Answer 1

Answer: (C)

$\left(x+^{n}C_{0}\right)\left(x+3. ^{n}C_{1}\right)\left(x+5. ^{n}C_{2}\right) .... \left(x+\left(2n+1\right).^{n}C_{n}\right)$
$= x^{n+1}+x^{n}\left\{^{n}C_{0}+3. ^{n}C_{1}+5. ^{n}C_{2}+..... +\left(2n+1\right).^{n}C_{n}\right\}+.....$
Coeff. of $x^{n} = ^{n}C_{0} +3. ^{n}C_{1}+5. ^{n}C_{2}+..... +\left(2n+1\right).^{n}C_{n}$
$ = 1+ \left(^{n}C_{1} +2. ^{n}C_{1}\right)+\left(^{n}C_{2}+4. ^{n}C_{2}\right) + ....+\left(^{n}C_{n}+2n. ^{n}C_{n}\right)$
$= \left(1+^{n}C_{1}+..... + ^{n}C_{n}\right)+2\left(^{n}C_{1} + 2^{n}C_{2} + .... +n. ^{n}C_{n}\right)$
$= 2^{n} +2\left[n+2. \frac{n\left(n-1\right)}{2!}+3. \frac{n\left(n-1\right)\left(n-2\right)}{3!}+...+ n.1\right]$
$= 2^{n} +2n \left[1+^{n-1}C_{1}+^{n-1}C_{2}+ ..... +^{n-1}C_{n-1}\right]$
$= 2^{n}+2^{n}. 2^{n-1} = 2^{n} \left(1+n\right) = \left(n+1\right) . 2^{n}$