So where
$ \hat n_{ML} = \text{max}_n ( \binom{n}{1000000} p^{1000000} (1-p)^{n-1000000} ) $
we can express x! as a gamma function but the derivative is kind of a mess. If you were to take the derivative of the the rest of the function you would find
$ {1000000}p^{999999} * -{n-1000000} *(1-p)^{n-1000001} $ = 0
