Maximum likelihood estimator for binomial model The main problem I'm having is that I'm getting $\hat{p}=\frac{\bar{x}}{n}$, not $\frac{x}{n}$.
For some reason, many of the derivations of the MLE for the binomial leave out the product and summation signs. When I do it without the product and summation signs, I get $\frac{x}{n}$, but leaving them in I get the following:
$$L=\prod_{i=1}^{n}{n \choose x_i}p^{x_i}(1-p)^{n-k_i}$$
$$\ell=\sum_{i=1}^{n}ln{n \choose x_i}+ln(p)\sum_{i=1}^{n}x_i  +ln(1-p)\sum_{i=1}^{n}(n-x_i)$$
$$\frac{\partial \ell}{\partial p}=\frac{\sum_{i=1}^{n} x_i}{p}-\frac{\sum_{i=1}^{n}(n-x_i)}{1-p}$$
$$\sum_{i=1}^{n}x_i-\sum_{i=1}^{n}x_ip=\sum_{i=1}^{n}(n-x_i)p$$
$$p=\frac{\sum_{i=1}^{n}x_i}{\sum_{i=1}^{n}(n-x_i)+\sum_{i=1}^{n}x_i}$$
$$p=\frac{\sum_{i=1}^{n}x}{n\cdot n}$$
$$p=\frac{\bar{x}}{n}$$
What am I doing wrong? Any help is appreciated.

You're confusing multiple binomial experiments with a single binomial experiment. Imagine you flip a coin 10 times and want to estimate the probability of Heads. If you observe 3 Heads, you predict $\hat{p} = \frac{3}{10}$. This is a sum of bernoullis, i.e. a single binomial experiment.
You're describing a sum of binomials, which corresponds to e.g. repeating your 10 flip experiment 5 times and observing:
$$X_1 = 3H$$
$$X_2 = 4H$$
$$X_3 = 2H$$
$$X_4 = 3H$$
$$X_5 = 4H$$
But if you think about it, doing 5 separate experiments in which you flip the coin 10 times is the same as doing a single experiment where you flip the coin 50 times. In this "condensed" experiment, we would estimate:
$$\hat{p} = \frac{16}{50} = \frac{\frac{16}{5}}{10} = \frac{\bar{x}}{n}$$
