Is it possible to simplify the following summations?
\begin{align} C&=\sum_{k=0}^N k\rho^k{N \choose k} \end{align}
and
\begin{align} P&= \sum_{k=0}^N k\rho^k P(N,k)= \\ &=\sum_{k=0}^N k\rho^k{N \choose k} k! \end{align}
Here, $C$ and $P$ stand for combination and permutation. Given that it easy to simplify the following expression, I was wondering if the corresponding expressions above are also amenable to simplification.
\begin{align} E&=\sum_{k=0}^N \rho^k{N \choose k} \\ &= (1+\rho)^N \end{align}
P.S.: if the expressions are not amenable to exact simplification, would there be a way to approximate them in closed form?
For $C$, absorb the $k$ via $k\binom{N}{k}=N\binom{N-1}{k-1}$, and then apply the binomial theorem to obtain $C=N \rho(1+\rho)^{N-1}$.