Question

The co-efficient of $x^k(0\le k\le n)$ in the expansion of $E=1+(1+x)+(1+x{)}^2+....+(1+x{)}^n$ is?

• A. ${}^{n+1}{C}_{k+1}$
• B. ${}^n{C}_k$
• C. ${}^{n+1}{C}_{n-k}$
• D. ${}^{n+1}{C}_{n-k-1}$

Answer 1

The correct option is A ${}^{n+1}{C}_{k+1}$

Given $E=1+(1+x)+(1+x{)}^2+..........+(1+x{)}^n$

It is a geometric progression with $(n+1)$ terms and ratio $(1+x)$

So, $E=\frac{(1+x{)}^{n+1}-1}{1+x-1}=\frac{(1+x{)}^{n+1}-1}{x}$

$=\frac{1+(n+2)x+..........+x^{n+1}-1}{x}$

$=(n+1)+n{+}^1{C}_{2}x+..........+x^n$

$\therefore$ The coefficient of $x^k$ in $E={}^{n+1}{C}_{K+1}$