Number of Permutations/Proof 1

Proof
We pick the elements of $S$ in any arbitrary order.

There are $n$ elements of $S$, so there are $n$ options for the first element.

Then there are $n - 1$ elements left in $S$ that we haven't picked, so there are $n-1$ options for the second element.

Then there are $n - 2$ elements left, so there are $n - 2$ options for the third element.

And so on, to the $r$th element of our selection: we now have $n - \paren {r - 1}$ possible choices.

Each mapping is independent of the choices made for all the other mappings, so by the Product Rule for Counting, the total number of ordered selections from $S$: