Probably a Constant

By rewriting and using the binomial theorem, $\sum_{k=0}^n \frac{p^k(1-p)^{n-k}n!}{k!(n-k)!} = \sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} = (p + (1-p))^n = 1^n = \boxed{1}$

Note: This actually holds for any real number $p \neq 0,1$ (and it will hold for $p=0,1$ as well if we define $0^0=1$ as is natural in discrete mathematics).

The summand part is the probability mass function (PMF) of Binomial Distribution with parameter $n$ and $p$ . So the sum of any PMF is $1$ .