The definition of a binomial point process that I'm working with is as follows: Let $S\subseteq\mathbb{R}^{d}$ with volume $|S|>0$, and let $f$ be a density function on $S$. A point process $X$ consisting of $n\in\mathbb{N}$ iid points with density $f$ is called a binomial point process.
My question: I'm having a hard time understanding why this is called a ''binomial'' point process. I've read somewhere that the ''binomial'' part follows from the fact that: $$ \mathbb{P}(N(B)=k)=\binom{n}{k}p^{k}(1-p)^{n-k}, $$ where $N(B)=\#(X\cap B)$ denotes the random number of points in $B\in\mathscr{B}(S)$, and $p=\int_{B}f(x)\,dx$. However, I can't seem to figure out how this follows from the definition above. Any help is greatly appreciated.
Short Answer. $N(B)$ counts the number of "successes" (where each point in $X$ is a success if it lies in $B$) in $n$ i.i.d. trials, hence it is binomially distributed.
Longer Answer. Let $(X_1, \ldots, X_n)$ be the $n$ i.i.d. points with density $f$. Then
$$ N(B) = \#(X \cap B) = \sum_{i=1}^{n} \#(\{X_i\} \cap B) = \sum_{i=1}^{n} \mathbf{1}_{\{X_i \in B\}}, $$
where $\mathbf{1}_{\{\ldots\}} $ is the indicator function notation. Since $X_i$'s are independent and each $\mathbf{1}_{\{X_i \in B\}}$ is a function of $X_i$, it follows that $\mathbf{1}_{\{X_i \in B\}}$'s are also independent. Moreover, each $\mathbf{1}_{\{X_i \in B\}}$ takes value
$$ \begin{cases} 1, & \text{with probability $p = \mathbf{P}(X_i \in B) = \int_B f(x) \, \mathrm{d}x$;} \\ 0 & \text{with probability $1-p$}; \end{cases} $$
hence it is a $\operatorname{Bernoulli}(p)$-variable. This tells that $N(B)$ is the sum of $n$ i.i.d. $\operatorname{Bernoulli}(p)$-variable, and therefore $N(B)$ has the binomial distribution with parameters $n$ and $p$.