Question

The coefficient of $x^k$ in the expansion of
$E=1+(1+x)+{(1+x)}^2+....+{(1+x)}^n$ is

• A. ${{}^{n}C}_k$
• B. ${{}^{n+1}C}_k$
• C. ${{}^{n+1}C}_{k+1}$
• D. none of these

Answer 1

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

Given, $E=1+(1+x)+(1+x{)}^2+...(1+x{)}^n$
$=\frac{1-(1+x{)}^{n+1}}{1-(1+x)}$
$=\frac{1}{x}\left[\right(1+x{)}^{n+1}-1]$
Hence, $x^k$ $=\frac{x^{k+1}}{x}$
Coefficient of $x^k$ will be
${}^{n+1}{C}_{k+1}$