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\title{MA162: Finite mathematics}
\subtitle{Permutations and Combinations}
\author{Paul Koester}
\institute{University of Kentucky}
\date{February 17, 2014}

\begin{document}

\maketitle

\begin{frame}[t]{6.4: Permutations}

\begin{itemize}
\item How many different ways can the letters O, P, and E be re-arranged?
\vfill
\item How many different ways can the letters in the word "MAINE" be re-arranged?
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Another Permutation}
\begin{itemize}
\item Bob, Jane, Mike, and Sally are elected to serve on a committee.  
\item One will have to serve as president, one as vice-president, one as treasurer, and one as secretary.
\item In how many ways can the positions be assigned to Bob, Jane, Mike, and Sally?
\vfill

\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Yet Another Permutation}
\begin{itemize}
  \item You have 100 songs in your i-Tunes library.
  \item How many ways can you create a playlist with 12 songs? \vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Permutations:  More formally}
\begin{itemize}
\item Given a set of distinct objects, a \alert{permutation} of this set is an arrangement 
      which specifies an ordering on the elements of the set. 
\vfill
\item The number of permutations of a set of n distinct items is 
\[
  n\cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1
\]
\vfill
\item The above expression is called \alert{n factorial} and is denoted \(n!\).  
\vfill

\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Permutations:  More formally}
\begin{itemize}
\item Given a set of distinct objects, a \alert{permutation} of this set is \alert{taken \(r\) at a time} is an arrangement 
      which chooses \(r\) of the elements of the set and specifies an ordering of these \(r\) items.. 
\vfill
\item The number of permutations of a set of n distinct items taken r at a time is 
\[
  P(n,r) = \frac{n!}{(n-r)!}
\]
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Permutations:  Not Necessarily Distinct Items}
\begin{itemize}
  \item You have a group of 3 boys and 4 girls.  
\vfill
  \item They are to stand in line. 
\vfill
  \item How many arrangements are there?  
\vfill

\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Permutations:  Not Necessarily Distinct Items}
\begin{itemize}
  \item You have a set with n items.  
\vfill
  \item \(n_1\) of the items are of alike of one kind.  \(n_2\) items are alike and of another kind.  ...
        \(n_r\) items are alike of yet another kind.
\vfill
  \item All of the items are one of the \(r\) types, so that 
        \[  n_1 + n_2 + \ldots + n_r = n \]  
\vfill
  \item The number of these n objects taken n at a time is then
        \[  \frac{n!}{n_1! n_2! \ldots n_r!} \]
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Permutations:  Rearranging Letters}
\begin{itemize}
  \item In how many ways can the letters in KENTUCKY be rearranged?
\vfill

  \item A poor musician has forgotten how to play her favorite melody.  
        She thought it was AACEGEA, but that doesn't sound right.  
        She knows that she has the right notes, and the right number of repeats of each note,
        She's just putting them in the wrong order.  In how many ways can she rearrange those notes? 
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Combinations - when order doesn't matter}
\begin{itemize}
  \item There are 10 new songs you'd like to download.  
\vfill
  \item You just received a gift card which will allow you to download 7 songs.
\vfill
  \item In how many ways can you choose 7 out of these 10 songs?  
\vfill
  \item NOTE:  The order in which you download the songs does not matter!
\vfill

\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Combinations - in general}
\begin{itemize}
  \item A \alert{combination} of n distinct items taken r at a time is an \alert{un-ordered selection} of r of these n items.
\vfill
  \item The number of such combinations is
        \[ \binom{n}{r} = C(n,r) = \frac{n!}{r!(n-r)!}\]
\vfill
  \item The expression \( \binom{n}{r} \) is pronounced ``n  choose  r.''  This is also called a ``binomial coefficient."
\vfill

\end{itemize}
\end{frame}

%\begin{frame}[t]{6.4: Combinations}
%\begin{itemize}
%  \item You walk home from school everyday.  
%\vfill
%  \item Home is 3 blocks south and 2 blocks east from school.
%\vfill
%  \item To keep your walk interesting, you try to take a different path every day. 
%\vfill
%  \item You also want to get home quickly, so you will never turn north, and you will never turn west.
%\vfill
%  \item How many different paths can you take?  
%\vfill
%
%\end{itemize}
%\end{frame}


\begin{frame}[t]{6.4: Counting Cards}
\begin{itemize}
  \item A standard deck of cards has 52 cards.   
\vfill
  \item A standard Poker hand has 5 of these cards.  
\vfill
  \item How many 5 card hands are possible?    
\vfill
  
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Counting Cards}
\begin{itemize}
  \item A standard deck of cards has the following cards: \\
A$\heartsuit$ 2$\heartsuit$ 3$\heartsuit$ 4$\heartsuit$ 5$\heartsuit$ 6$\heartsuit$ 7$\heartsuit$ 8$\heartsuit$ 9$\heartsuit$ 10$\heartsuit$ J$\heartsuit$ Q$\heartsuit$ K$\heartsuit$ \\
  A$\diamondsuit$ 2$\diamondsuit$ 3$\diamondsuit$ 4$\diamondsuit$ 5$\diamondsuit$ 6$\diamondsuit$ 7$\diamondsuit$ 8$\diamondsuit$ 9$\diamondsuit$ 10$\diamondsuit$ J$\diamondsuit$ Q$\diamondsuit$ K$\diamondsuit$ \\
  A$\clubsuit$ 2$\clubsuit$ 3$\clubsuit$ 4$\clubsuit$ 5$\clubsuit$ 6$\clubsuit$ 7$\clubsuit$ 8$\clubsuit$ 9$\clubsuit$ 10$\clubsuit$ J$\clubsuit$ Q$\clubsuit$ K$\clubsuit$ \\
  A$\spadesuit$ 2$\spadesuit$ 3$\spadesuit$ 4$\spadesuit$ 5$\spadesuit$ 6$\spadesuit$ 7$\spadesuit$ 8$\spadesuit$ 9$\spadesuit$ 10$\spadesuit$ J$\spadesuit$ Q$\spadesuit$ K$\spadesuit$ \\

  \item See \url{http://en.wikipedia.org/wiki/List_of_poker_hands} for definitions of poker hands.
\vfill
  \item Try counting the number of each type of poker hand yourself.  This is a great way to practice the ideas from this chapter.
\vfill
  \item Check your work against wikipedia's, but be warned that not everything you read on \emph{wikipedia} is reliable!  
\vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{6.4: Full house or 4 of a kind?}
\begin{itemize}
  \item A \alert{full house} is a hand with 3 of one kind and 2 of another kind. 
\vfill
  \item A \alert{four of a kind} is a hand with 4 of one kind, and some other card.
\vfill
  \item Which is more common?  Full house or 4 of a kind?    
\vfill
\end{itemize}
\end{frame}

%\begin{frame}[t]{6.4: Two pair or 3 of a kind?}
%\begin{itemize}
%  \item A \alert{Two pair} consists of two cards of one kind, two cards of different kind, and a fifth card, whose kind does not match either of the pairs.
%\vfill
%  \item For example, A$\heartsuit$ A$\spadesuit$ 5$\clubsuit$ 5$\heartsuit$ 7$\heartsuit$ 
%\vfill
%  \item A \alert{Three of a kind} consists of three cards of one kind, and two other cards, in which the other two cards do not form a pair, nor do either of 
%        kinds match the three of a kind.  
%\vfill
%  \item For example, A$\heartsuit$ A$\spadesuit$ A$\diamondsuit$ 5$\clubsuit$ 7$\spadesuit$ 
%\vfill
%  \item Which is more common?  Two pair or 3 of a kind?  
%\vfill
%\end{itemize}
%\end{frame}
%
%
%
%\begin{frame}[t]{6.4: Flush or Straight?}
%\begin{itemize}
%
%  \item A \alert{full house} is a hand with 3 of one kind and 2 of another kind. 
%\vfill
%  \item A \alert{straight} is a hand with 5 cards in sequence, but not all of the same suit.
%\vfill
%  \item For example, 8$\heartsuit$ 9$\spadesuit$ 10$\clubsuit$ J$\heartsuit$ 7Q$\heartsuit$ 
%        Note:  an ACE can go before 1 or after King.  
%\vfill
%  \item Which is more common?  Full house or a straight (but not straight flush)?  
%\vfill
%  
%\end{itemize}
%\end{frame}



\begin{frame}[t]{6.4: Counting Cards}
\begin{itemize}
  \item A hand in 7 card stud consists of 7 cards, in which 3 cards are face-up (visisble to all players)
        and 4 cards are face-down.
\vfill
  \item How many distinct hands can be dealt to a player in 7 card stud?  
\vfill
  \item NOTE:  It is NOT \( \binom{52}{7} \).
\vfill

\end{itemize}
\end{frame}

\end{document}

