How to compute the summation of a binomial coefficient/ show the following is true $\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$.
I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make the binomial coefficient useful.

Let $X$ be the number of heads in $n$ flips of a fair coin (i.e. a random variable with $\text{Binomial}(n,1/2)$ distribution).  Then $$\sum_{k=0}^ n {n
\choose k} 2^{-n} (2k+1) = \mathbb E[2X+1] = 2 \mathbb E[X] + 1 = n + 1$$
