Construction of Permutations/Proof 1

Proof
The following is an inductive method of creating all the permutations of $n$ objects.

Base Case
There is clearly one way to arrange one object in order.

Inductive Hypothesis
We assume that we have constructed all $n!$ permutations of $n$ objects.

Induction Step
, let a set $S_n$ of $n$ objects be $\set {1, 2, \ldots, n}$.

Take a permutation of $S_n$:
 * $a_1 \, a_2 \, a_3 \, \ldots \, a_n$

Now we take the number $n + 1$.

We can form $n + 1$ permutations from this one by putting $n + 1$ in all places possible:
 * $a_{n + 1} \, a_1 \, a_2 \, a_3 \, \ldots \, a_n, \quad a_1 \, a_{n + 1} \, a_2 \, a_3 \, \ldots \, a_n, \quad a_1 \, a_2 \, a_{n + 1} \, a_3 \, \ldots \, a_n, \quad \ldots, \quad a_1 \, a_2 \, a_3 \, \ldots \, a_n \, a_{n + 1}$

It is clear that all permutations of $n + 1$ objects can be obtained in this manner, and no permutation is obtained more than once.

As there are ${}^n P_n$ permutations on $n$ objects, there are $\paren {n + 1} \times {}^n P_n$ permutations on $n + 1$ objects.

Hence by induction, and the recursive definition of the factorial:
 * ${}^n P_n = n!$