Solving some problem I have stumbled into the following sum :
$ \displaystyle \sum_{i=0}^{n-2} {e \choose i} (n-1-i) (1-p)^{e-i} p^i $
where $ 0 \leq e \leq {n \choose 2}$.
I am not very efficient with the evaluation of these sums so I would like to ask if there is any way to evaluate this sum or obtain a sharp lower bound for it?