Consider the following summation:
$$\sum_{k=0}^n k\binom{n}{k}(1-p)^{n-k}p^k$$
If the $k$ term was not present it would be a simple binomial. However, because of the $k$ term I am unable to derive the sum.
I believe the summation should be equal to $np$ (The summation is part of a larger expression, which upon solving using a different approach gives the $np$ result). I have tested this for smaller values of $n = \{1,2,3\}$.