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\centerline{\bf Homework 11, MORALLY Due 10:00AM May 6} 
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\begin{enumerate}


\item
(30 points- 5 points each) 
After Emily gets her PhD she works for a casino. 
She has the following dyslexic idea:
Rather than have 13 ranks and 4 suites, lets have
13 suites and 4 ranks!
She also changes the number of cards in a hand to 4
so that a straight is possible. 

\begin{enumerate}
\item
How many poker hands are there?
\item
What is the probability of getting a straight flush?
\item
What is the probability of getting a straight that is not a straight flush?
\item
What is the probability of getting a flush that is not a straight flush?
\item
What is the probability of getting 4-of-a-kind?
\item
What is the probability of getting 3-of-a-kind that is NOT 4-of-a-kind? 
\end{enumerate} 


\newpage


\item 
(25 points)
Bill is looking at three dice. 
Bill will pick up one die and roll it. 
\begin{itemize}
\item
Dice 1 has 3 sides labelled 1,2,3.  The probabilities are: 

prob(1)=0.3, prob(2)=0.3, prob(3)=0.4. 

Bill picks up this die with probability 0.5
\item
Dice 2 has 4 sides labelled 1,2,3,4. The probabilities are: 

prob(1)=0.3, prob(2)=0.3, prob(3)=0.2, prob(4)=0.2.

Bill picks up this die with probability 0.3
\item
Dice 3 has 5 sides labelled 1,2,3,4,5. The probabilities are:

prob(1)=prob(2)=prob(3)=prob(4)=prob(5)=0.2.

Bill picks up this die with probability 0.2
\end{itemize}

\begin{enumerate}
\item 
(10 points) 
What is the probability that Bill rolls a 3?
Show your work.
\item
(15 points) 
(You will probably want to write a program for this one.)
Give a table like the one below (the numbers in it are WRONG, yours should be right).

\[
\begin{array}{|c|c|}
\hline
\hbox{Number} & \hbox{Prob} \cr 
\hline
1             & 0.3         \cr
2             & 0.2         \cr
3             & 0.1         \cr
4             & 0.2         \cr
5             & 0.2         \cr
\hline
\end{array}
\] 

\end{enumerate} 

\newpage

\item
(20 points) 
(This problem was inspired by a comment Soren made.) 

Let $E(k,n)$ be the number of solutions in $\N$ to

$$x_1 + \cdots + x_k = n.$$

In class we showed that $E(k,n)=\binom{n+k-1}{n}$.

\begin{enumerate}
\item
(10 points) 
Pretend that you {\it do not know} the formula for $E(k,n)$. 
But you (actually Soren) have the following idea!

EITHER $x_k=0$ OR $x_k=1$ OR $x_k=2$ OR $\cdots$ $x_k=n$. 

Use this to get a recurrence for $E(k,n)$. 
Also make sure to have a base case (perhaps more than one)
that makes sense. 


\item
(10 points)
From Part 1 you have a recurrence for $E(k,n)$.
Now use the fact that you DO know $E(k,n)=\binom{n+k-1}{n}$ to get
a combinatorial identity. 

\newpage


\end{enumerate}

\newpage

\item
(25 points) 
Fill in the $f(c)$ below and then prove your statement:

For any $c$-coloring of the $(c+1)\times f(c)$ grid there is a monochromatic
rectangle.

No proof needed though you may want to do one for your own benefit. 

\end{enumerate}

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