For atomic orbitals:
E2 orb: $\binom{N-n_1}{n_2}$
E2 orb: $\binom{N-n_1-n_2}{n_3}$
E2 orb: $\binom{N-n_1-n_2-n_3}{n_4}$
…
En orb: $\binom{n_i}{n_i}$
now probability function is:
$P= N! \prod^{n}_{n=1}\frac{1}{n_{i}!}$
Why? In general?
[Update]
Every combination is greater than 1. So their product is greater than 1. How on earth can such multiplication lead to a probability function? Is the probability function scaled back to range $[0,1]$?
Best Answer
As you correctly point out, the terminology on page 10 of the lecture notes you linked to is incorrect: $P$ is the number of microstates making up the given macrostate; it is not a probability, since it typically exceeds one.
However, dividing $P$ by the total number of microstates $n^N$ does give the probability of the given macrostate, under the assumption that all microstates are a priori equally likely. Since the constant $1/n^N$ is the same for all macrostates of the system, we may safely ignore it, as the author of the lecture notes does, when comparing the probabilities of different macrostates.
(If I were reviewing those notes, I'd also point out that the author seems rather excessively fond of the letter "n". Having $n$ and $N$ as system parameters is confusing enough, but when he then introduces the macrostate parameters $n_1$, $n_2$ and up to $n_n$...)