\begin{defn}
We define the factorial of an integer n as
$$
n!= n\cdot(n-1) \cdot(n-2)\cdots \ldots \cdot 3 \cdot 2 \cdot 1 .
$$
\end{defn}

\begin{xmpl}

Suppose you have 6 apples, $\{a, b, c, d, e, f\}$ and you
want to put each one into a different apple basket, $\{1,2,3,4,5,6\}$.\\

For the first basket you can choose from 6 apples $\{a, b, c, d, e,
f\}$, and for the second basket you have then 5 apples to choose from
and so it goes for the rest of the baskets, so for the last one you
only have 1 apple to choose from.\\

The end result would then be:
$6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$ possible allocations.\\

This could also be calculated in R with the factorial function:
\begin{lstlisting}
factorial(6)
[1] 720
\end{lstlisting}
\end{xmpl}