Combinatorial numbers

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

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 $m⩾n$.

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.

The formula that allows us to find the value of a combinatorial number is the following one: $$\left(\begin{array}{c}m\\ n\end{array}\right)={\displaystyle \frac{m!}{n!\cdot (m-n)!}}$$

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

$$\left(\begin{array}{c}a\\ a\end{array}\right)={\displaystyle \frac{a!}{a!(a-a)!}}={\displaystyle \frac{a!}{a!\cdot 0!}}=1$$

It is also easy to verify that $\left(\begin{array}{c}n\\ 0\end{array}\right)=1$. $$\left(\begin{array}{c}n\\ 0\end{array}\right)={\displaystyle \frac{n!}{0!(n-0)!}}={\displaystyle \frac{n!}{n!}}=1$$

And also that $\left(\begin{array}{c}m\\ 1\end{array}\right)=m$. $$\left(\begin{array}{c}m\\ 1\end{array}\right)={\displaystyle \frac{m!}{1!(m-1)!}}={\displaystyle \frac{m\cdot \cancel{(m-1)!}}{\cancel{(m-1)!}}}=m$$

Therefore to calculate combinatorial numbers we must remember the following formulas: $$\left(\begin{array}{c}m\\ n\end{array}\right)={\displaystyle \frac{m!}{n!\cdot (m-n)!}}\left(\begin{array}{c}n\\ n\end{array}\right)=1\left(\begin{array}{c}n\\ 0\end{array}\right)=1\left(\begin{array}{c}n\\ 1\end{array}\right)=n$$