I've seen several relatively length derivations of $E[X], \text{var}(X)$, where $X \sim binomial$. Actually, in fact, all the resources seem to use the lengthy approach.
The lengthier approach is doing something like: \begin{align} E[X] = \sum_{x=0}^n x \binom{n}{x} p^x (1 - p)^{n - x} \\ = \sum_{x=0}^n x \frac{n!}{(n-x)!x!} p^x (1 - p)^{n - x} \\ = \sum_{x=1}^n x \frac{n!}{(n-x)!x!} p^x (1 - p)^{n - x} \\ = np\sum_{x=1}^n \frac{(n-1)!}{(n-x)!(x-1)!} p^{x-1} (1 - p)^{n - x} \\ = np\sum_{x=0}^{n-1} \frac{(n-1)!}{(n-x-1)!(x)!} p^{x} (1 - p)^{n - x - 1} \\ = np \sum_{x=0}^{n-1} \binom{n-1}{x}p^x(1 - p)^{n - x - 1} \\ = np (p + 1 - p)^{n-1} \\ = np \end{align}
and analogously for $E[X^2]$ to get the eventual variance.
The really simply approach, and I'm not sure if it is fully valid, is to just introduce $B_i$ as a bernoulli random variable where $B_i = 1$ with probability $p$.
Then, easily,
\begin{align} E[X] = E[B_1 + \ldots B_n] = nE[B_1] = np \end{align}
Is this approach valid or is it flawed in some way? The one thing I am concerned about is in the former lengthier approach, we consider all possibly values of $X$ and their associated probabilities. Am I really doing that with this bernoulli/indicator variable approach? If not, does this make this approach invalid?