Combinatorial numbers

A combinatorial number is formed by two positive integers $$m$$ and $$n$$ written one on top of the other, within brackets: $$$ \begin{pmatrix} m \\ n \end{pmatrix}$$$

There is only one thing to bear in mind, apart from the fact that $$m$$ and $$n$$ must be positive integers: the top number cannot be smaller than the bottom one, that is, it must always be $$m\geqslant n$$.

To write them we will use the matrix tab in the formula editor, so that we will always insert a matrix with 1 column and 2 rows.

These are combinatorial numbers: $$$ \begin{pmatrix} 5 \\ 2 \end{pmatrix}, \quad \begin{pmatrix} 24 \\ 7 \end{pmatrix}, \quad \begin{pmatrix} 45 \\ 23 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$$

The formula that allows us to find the value of a combinatorial number is the following one: $$$\begin{pmatrix} m \\ n \end{pmatrix}=\dfrac{m!}{n!\cdot(m-n)!}$$$

Let's see some examples:

$$ \begin{pmatrix} 5 \\ 2 \end{pmatrix}=\dfrac{5!}{2!\cdot(5-2)!}= \dfrac{5\cdot4\cancel{3\cdot2\cdot1}}{2\cdot1\cancel{3\cdot2\cdot1}}= \dfrac{20}{2}=10$$

$$ \begin{pmatrix} 4 \\ 3 \end{pmatrix}=\dfrac{4!}{3!\cdot(4-3)!}= \dfrac{4\cancel{3\cdot2\cdot1}}{\cancel{3\cdot2\cdot1}\cdot1}=4$$

$$ \begin{pmatrix} 4 \\ 4 \end{pmatrix}=\dfrac{4!}{4!\cdot0!}= 1$$

In the last example we have used $$0! = 1$$.

In general we will always obtain the value $$1$$ when both numbers are equal, that is to say that $$\begin{pmatrix} a \\ a \end{pmatrix}=1$$ since:

$$$\begin{pmatrix} a \\ a \end{pmatrix}= \dfrac{a!}{a!(a-a)!}=\dfrac{a!}{a!\cdot0!}=1$$$

It is also easy to verify that $$\begin{pmatrix} n \\ 0 \end{pmatrix}=1$$. $$$\begin{pmatrix} n \\ 0 \end{pmatrix}= \dfrac{n!}{0!(n-0)!}=\dfrac{n!}{n!}=1$$$

And also that $$\begin{pmatrix} m \\ 1 \end{pmatrix}=m$$. $$$\begin{pmatrix} m \\ 1 \end{pmatrix}= \dfrac{m!}{1!(m-1)!}= \dfrac{m\cdot\cancel{(m-1)!}}{\cancel{(m-1)!}}=m$$$

Therefore to calculate combinatorial numbers we must remember the following formulas: $$$\begin{pmatrix} m \\ n \end{pmatrix}=\dfrac{m!}{n!\cdot(m-n)!} \quad \begin{pmatrix} n \\ n \end{pmatrix}=1 \quad \begin{pmatrix} n \\ 0 \end{pmatrix}=1 \quad \begin{pmatrix} n \\ 1 \end{pmatrix}=n $$$