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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Permutations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Permutations\label{Permutations}}

A \textbf{permutation} of a set of distinct objects is an ordering of these
objects. For example, some of the permutations of the letters \emph{ABCDWXYZ}
are

\begin{center}
\begin{tabular}{lllllll}
\textit{XAYBZWCD} &  & \textit{ZAYBCDWX} &  & \textit{DBWAZXCY} &  & \textit{%
YDXAWCZB}
\end{tabular}
\end{center}

\noindent How many such permutations are possible? Since there are eight
choices for the first position, seven for the second (after the first has
been chosen), six for the third (after the first two have been chosen), and
so on, the Fundamental Counting Principle tells us that the number of
possible permutations is 
\[
8\times 7\times 6\times 5\times 4\times 3\times 2\times 1=40,320
\]

This same reasoning with $8$ replaced by $n$ leads to the following
observation:

\begin{center}
\label{Permutations of n objects}The number of permutations of $n$ objects
is $n!$\bigskip 
\end{center}

\noindent In general, if a set has $n$ elements, then the number of ways of
ordering $r$ elements from the set is denoted by $P(n,r)$ and is called 
\textbf{the} \textbf{number of permutations} of $n$ \textbf{objects taken} $r
$ \textbf{at a time.}

\subsection{Permutations of\ n\ Objects Taken\ r\ At a Time\label%
{Permutations of n objects taken r at a time}}

The number of permutations of $n$ objects taken $r$ at a time is 
\[
P(n,r)=\frac{n!}{\left( n-r\right) !}
\]

\subsection{Distinguishable Permutations\label{Distinguishable permutations}}

If a set of $n$ objects consists of $k$ different kinds of objects with $%
n_{1}$ objects of the first kind, $n_{2}$ objects of the second kind, $n_{3}$
objects of the third kind, and so on, where $n_{1}+n_{2}+\cdots +n_{k}=n$,
then the number of distinguishable permutations of these objects is 
\[
\frac{n!}{n_{1}!n_{2}!n_{3}!\cdots n_{k}!}
\]

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Factorial}{}{}{../../Help/DM2-2.tex#Factorial}}

\noindent \hrulefill

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\noindent \hrulefill

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