Order of Symmetric Group

Theorem
Let $$S_n$$ be the symmetric group on $n$ letters.

Then $$S_n$$ has $$n!$$ elements (see factorial).

Proof
A direct application of Cardinality of Set of Bijections.

Example
Thus, when $$n = 3$$, there are $$3 \times 2 \times 1 = 6$$ permutations:

$$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{bmatrix} \qquad \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{bmatrix} \qquad \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{bmatrix}$$

$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix} \qquad \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{bmatrix} \qquad \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{bmatrix}$$