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Counting project \hfill MA162\\
\semester \hfill Russell Brown

\smallskip

Consider the problem of counting the number of outcomes of a complex
operation. This hard counting problem is solved by breaking the
complex operation up into simpler steps.  If we can give the number of
outcomes for each simpler step, then we use the multiplication
principle to compute the number of outcomes of the complex operation
as the product of the number of outcomes for each of the simpler
steps.

One of the most important applications of the multiplication principle is the
derivation of the formula for the number of permutations of $n$ things
taken $k$ at a time: $P(n,k ) = n!/(n-k)!$.
Let us recall how to do this  by an example.

\begin{example} How many three letter words can we form from
a set of ten letters if repeated letters are not allowed.  
\end{example}

\begin{solution} We break
the complex operation  of choosing three letters into the  simpler
operations of choosing individual letters.  The first letter may be
chosen in 10 ways, the second in 9 ways, the  third in 8 ways. After
each choice, the number of letters available decreases by one because
we are not allowing repeated letters.  Thus
the total number of words is $ 10 \times 9 \times 8 = 720$. Of course,
this agrees with the formula $P(10,3 ) = 10!/7!$.  
\end{solution}

In this project,  we will apply this strategy to counting poker
hands.  In each case, our goal is to find a sequence of operations
which produce all the hands of a given type and do not produce a hand
more than once.   In all of the problems below, assume that you are
working with a standard deck of 52 cards. This deck is divided into
four suits, and each suit has 13 ``numbers'' 2-10, J, Q, K, A.
A poker hand contains five cards.  Thus there are a total of
$C(52,5)=2,598,960$ 
hands.

\begin{example}  How many hands contain two pairs?  
\end{example}

\begin{solution} We make a table listing one sequence of operations
that will produce all poker hands with two pairs.

\begin{tabular}{l|l}
Operation & Number of outcomes \\
\hline
Choose 2 of 13 suits  & C(13,2)\\
Choose 2 of 4 cards  in first suit & C(4,2)\\
Choose 2 of 4 cards in second suit & C(4,2) \\
Choose 1 of the 44 cards which are not in the chosen suits & C(44,1)\\
\hline
TOTAL & 123,552
\end{tabular}

Notice that by choosing the suits first and then choosing the last
card from the remaining 44 cards, we guarantee that we do not draw a
four of a kind or a full house.
\end{solution}

Now, we try a much easier example.
\begin{example} How many royal flushes are there?  A royal flush is
the cards 10, J, Q, K, A in the same suit.
\end{example}

\begin{solution}
Four. Once you choose the suit, the hand is completely determined.
\end{solution}

Now practice applying the multiplication principle to the problems
below.  In your answers, you should give the simple operations that
you carry out to produce all the hands of a specified type.  Try to
explain any particular tricky points. 

\begin{enumerate}
\item Count the number of full houses.  A full house consits of two
cards from one number and 
three from another.
\item Count the number of hand which contain a pair.
Do not include hands which contain three of a kind or two pairs.
\item Count the number of straight flushes.  A straight flush consists
of five cards in order from the same suit.  For this problem, you may
assume that the ace, A, may 
be high or low. 

\item  Count the number of straights which are not straight flushes.
A straight consists of five 
cards in order.  

\item Count the number of flushes which are not straight flushes.  A
flush consists of five cards 
from the same suit. 
\end{enumerate}

Work in a group of 2 to 5 to answer these questions.  
After you have answered these questions, write up your solutions in an
orderly fashion.  Turn in your solutions on  \duedate\  at the beginning
of lecture. You should include the names and section numbers of   each
member of your group.  Note that the answers to most of these
questions may be found in the text or in a book about card games. In
the text, look at problems 50--59 in section 5.3 and at the table 
on page 324.
Since you know the answers to these questions, your grade will be
determined, primarily, by  how clearly you present your solution and
explain your 
reasoning. 


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{\small \today}
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