二項係數
binomial coefficients$ \begin{pmatrix}n \\ k\end{pmatrix},$ _nC_k
$ \begin{pmatrix}n \\ k\end{pmatrix}:=\frac{n!}{k!(n-k)!}=\frac{(n)_k}{k!}=\frac{(n)_k}{(k)_k}.
漸化式$ \begin{pmatrix}n \\ k\end{pmatrix}=\begin{pmatrix}n-1 \\ k-1\end{pmatrix}+\begin{pmatrix}n-1 \\ k\end{pmatrix},$ \begin{pmatrix}n \\ n\end{pmatrix}=\begin{pmatrix}n \\ 0\end{pmatrix}=1
$ \begin{pmatrix}n \\ k\end{pmatrix}=\begin{pmatrix}n \\ n-k\end{pmatrix}.
二項定理
$ (x+y)^n=\sum^n_{k=0}\left(\begin{matrix}n \\ k\end{matrix}\right)x^{n-k}y^k=\sum^n_{k=0}\left(\begin{matrix}n \\ k\end{matrix}\right)x^ky^{n-k}.
$ (1+x)^n=\sum^n_{k=0}\left(\begin{matrix}n \\ k\end{matrix}\right)x^k.
$ (x+y)^a=\sum^\infty_{k=0}\left(\begin{matrix}a \\ k\end{matrix}\right)x^{a-k}y^k,$ a\in\Complex
二項級數 (binomial series)$ \sum^\infty_{k=0}\left(\begin{matrix}a \\ k\end{matrix}\right)z^k
$ (1-z)^{-a}=\sum^\infty_{n=0}(a)^n\frac{z^n}{n!}=~_1F_0\left(\begin{matrix}a \\ -\end{matrix};z\right).
$ (1+z)^a=~_1F_0\left(\begin{matrix}-a \\ -\end{matrix};-z\right)=\sum^\infty_{k=0}\left(\begin{matrix}a \\ k\end{matrix}\right)z^k.
q-二項係數$ \left(\begin{matrix}n \\ k\end{matrix}\right)_q 二項分布 (binomial distribution)$ B(n,p)
多項係數 (multinomial coefficient)$ \left(\begin{matrix}n \\ k_1,k_2,…,k_r\end{matrix}\right)
$ \left(\begin{matrix}n \\ k_1,k_2,…,k_r\end{matrix}\right):=\frac{n!}{k_1!k_2!,…,k_r!}.
多項分布 (multinomial distribution)