how comes this step ? inductive proof with binomial

Question

I am sitting on this problem for couple of hours now but still cannot get the reason why the step which i marked is possible...

how has $(n+1)$ disappeared here?

Answer 1

Let's write it with an indication of what's what:

$$\begin{multline} (\underbrace{n}_A + \underbrace{1}_B) + \underbrace{\sum_{k=1}^n k\binom{n}{k}}_C + \underbrace{\sum_{k=1}^n (k-1)\binom{n}{k-1}}_D + \underbrace{\sum_{k=1}^n\binom{n}{k-1}}_E\\ = \underbrace{\sum_{k=1}^n k\binom{n}{k}}_C + \Biggl(\underbrace{\sum_{k=0}^{n-1}k\binom{n}{k}}_D + \underbrace{n}_A\Biggr) + \Biggl(\underbrace{\sum_{k=0}^{n-1} \binom{n}{k}}_E + \underbrace{1}_B\Biggr) \end{multline}$$

Note that in the sums $D$ and $E$, the index $k$ was shifted too, which makes it harder to see what's what.