\begin{align*}
\mu & = E[X] \\
& = \sum_{x=0}^{n} {x \binom{n}{x} p^x (1-p)^{n-x}}\\
& = \sum_{x=1}^{n} {x \frac{n(n-1)!}{x(x-1)!(n-x)!} p^x (1-p)^{n-x}}\\
& = np \sum_{x=1}^{n} {\frac{(n-1)!}{(x-1)!((n-1)-(x-1))!} p^{x-1} (1-p)^{(n-1)-(x-1)}}
\end{align*}
Using the change of variables \(k=x-1\) and \(m = n-1\) yields a binomial series
\begin{align*}
& = np \sum_{k=0}^{m} {\frac{m!}{k!(m-k)!} p^k (1-p)^{m-k}}\\
& = np (p + (1-p))^m = np
\end{align*}