Theorem 6.2. The order of $S_n$ is $n!$.
PROOF. The problem of computing the number of elements in $S_n$ is the same as that
of computing the number of different ways the integers 1, 2,..., n can be placed in the n
blanks indicated (using each integer just once):
\begin{pmatrix}
1 & 2 & \cdots & n \\
\_ & \_ & \cdots & \_
\end{pmatrix}
If we begin filling these blanks from the left, there are n possibilities for the first blank.
Once that choice has been made, there remain $n - 1$ possibilities for the second blank.
Then there are $n - 2$ possibilities for the third blank, and so on. The theorem follows by
repeated application of this basic counting principle: If one thing can be done in r different
ways, and after that a second thing can be done in s different ways, then the two things can
be done together in rs different ways.
