On line 3, if you multiply $$L(p \mid \boldsymbol y) = f(\boldsymbol y \mid p) = \prod_{i=1}^n \binom{n}{y_i} p^{y_i} (1-p)^{n-y_i}.$$
why does the power on the (1-p) not go to $n^2 - \sum_{i=1}^n Y_i$ since youre multiplying (n-yi) n times?
Thank you!
You're right, it should be $n^2$.
(It's not clear to me why you have $n$ both as the number of trials in the binomial distribution and as the number of data points, but that doesn't invalidate your argument; the exponent should contain the product of the two.)