TITLE: Simplify $\sum_{i=0}^n (i+1)\binom ni$
QUESTION [6 upvotes]: Simplifying this expression$$1\cdot\binom{n}{0}+ 2\cdot\binom{n}{1}+3\cdot\binom{n}{2}+ \cdots+(n+1)\cdot\binom{n}{n}= ?$$
$$\text{Hint: } \binom{n}{k}= \frac{n}{k}\cdot\binom{n-1}{k-1} $$

REPLY [2 votes]: $$\sum_{i=0}^n (i+1)\binom ni=\sum_{i=0}^n i\binom ni+\sum_{i=0}^n \binom ni=\sum_{i=0}^n i\frac ni\binom {n-1}{i-1}+2^n=$$
$$=n\sum_{i=0}^n\binom {n-1}{i-1}+2^n=n\sum_{j=0}^n\binom {n-1}{j}+2^n=n2^{n-1}+2^n$$