Binomial Theorem Question $1+(1+x)+(1+x)^2+\dots+(1+x)^n$

Question

The coefficient of $x^k (0 \le k \le n)$ in the expansion of $E = 1+(1+x)+(1+x)^2+\dots+(1+x)^n$ is

\begin{alignat}{2} \text{(A)}& \binom{n+1}{k+1} &\qquad \text{(B)}& \binom{n}{k} \\ \text{(C)}& \binom{n+1}{n-k-1} &\qquad \text{(B)}& \binom{n+1}{n-k-1} \end{alignat}

Please help me or provide an useful hint for the question.

Answer 1

Since hint should be posted as answer, I post my hints here.

\begin{align}\sum_{r = 0}^n (1+x)^r&=\frac{(1+x)^{n+1}-1}{(1+x)-1} \\ &=\frac1x\left[\sum_{k=0}^{n+1} \binom{n+1}{k}x^k\right] - \frac1x \end{align}