
%\bigskip
%\noindent\textbf{\large Combinatorics}
\section{Combinatorics}

$n$ items, pick $k$ of them such that order matters and any object
may be used more than once.  Number of ways is $n^k$.

$n$ items, pick $k$ of them such that order matters, but objects
may only be used once.  Number of ways 
\begin{displaymath}
  (n)_k = \frac{n!}{(n - k)!}
\end{displaymath}

$n$ items, pick $k$ of them such that order does not matter and each object
can be used only once.  Number of different sets we can have is

\begin{displaymath}
  (n)_k / k! = {n\choose k} = \frac{n!}{k! (n - k)!} = {n \choose {n - k}}
\end{displaymath}

$n$ items, pick $k$ of them such that order does not matter and each
object can be used more than once.  Number of different sent we can have is

\begin{displaymath}
  \frac{(n + k - 1)!}{k! (n - 1)!} = {n + k - 1 \choose k} = {n + k - 1 \choose n - 1}
\end{displaymath}

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