TITLE: Probability notation two numbers stacked inside brackets
QUESTION [16 upvotes]: I have a very simple question. Could someone elaborate on the notation of two numbers stacked inside brackets? Such as:
$$
        \begin{pmatrix}
        5 \\
        1 \\
        \end{pmatrix}
$$

REPLY [2 votes]: The most valuable definition is the following:
$$\binom{z}{k} = \begin{cases}\frac{z(z - 1)\cdots(z - k + 1)}{k!} = \frac{z^{\frac{k}{}}}{k!} & \text{for integer $k \ge 0$},\\
0 &\text{for integer $k < 0$,}\end{cases}$$
where $z$ is a complex number. See textbook "Concrete Mathematics: A Foundation for Computer Science" by Ronald Graham, Donald Knuth, and Oren Patashnik.
It is easy to see that $\binom{n}{k} = \frac{n!}{(n - k)!k!}$ for $0 \le k \le n$ that is the number of choices of $k$ objects from $n$ objects. But more interesting is the following:
$$(1 + x)^{\alpha} = \sum_{k = 0}^{\infty}\binom{\alpha}{k}x^k$$
for any real $\alpha$.