Probability notation two numbers stacked inside brackets I have a very simple question. Could someone elaborate on the notation of two numbers stacked inside brackets? Such as:
$$
        \begin{pmatrix}
        5 \\
        1 \\
        \end{pmatrix}
$$
 A: This is the usual notation for a binomial coefficient: $\binom{123}{45}$ is the number of different $45$-element subsets of a set with $123$ elements in total. It can be computed as
$$ \binom{123}{45} = \frac{123!}{45!(123-45)!} $$
In some elementary probability/combinatorics texts, the same concept can also be notated something like ${}^{123}C_{45}$ or ${}_{123}C_{45}$ or $C(123,45)$.
A: It's the 'choose' function. Say you have 8 people and you want to pick (choose) 3 of them to form a team. Then the number of different teams you can create is $8\choose3$.
A: Assume $m\geq n$ are nonnegative integers. We define $\binom{m}{n}$ as 
$$ 
\binom{m}{n} := \frac{m!}{n!\,(m-n)!} = \dfrac{m(m-1)(m-2)\cdots 3\cdot 2\cdot 1}{(n(n-1)\cdots 3\cdot 2\cdot 1)((m-n)(m-n-1)\cdots 3\cdot 2\cdot 1)},  
$$ 
and read $\binom{m}{n}$ as "$m$ choose $n$". As others have mentioned above, this is called the $\textbf{binomial coefficient}$. 
Let's go back to the example $\binom{5}{1}$. One could think of this as the number of ways to choose $1$ object in a bag of $5$ objects. If you have a bag with $5$ objects, how many ways are there to pick one item? There are $5$ ways. 
That is, 
$$
\binom{5}{1}=\frac{5!}{1! (5-1)!} = \frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{1\cdot 4\cdot 3\cdot 2\cdot 1} = 5.
$$
Now consider the example $\binom{5}{2}$. How many ways are there to pick two items in a bag with $5$ items (without replacement)? 
There are $5$ ways to pick the first item (let's call this item $A$) and then there are $4$ ways to pick the second item (let's call this item $B$). Since the ordering does not matter, we divide $5\cdot 4$ by $2$ to obtain: 
$$
\frac{5\cdot 4}{2} = 10,  
$$ 
which is 
$$
\binom{5}{2} = \frac{5!}{2!\, 3!} = \frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1\cdot 3\cdot 2\cdot 1} = 10.  
$$
A: 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 $\alpha$.
