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\noindent{\color{blue}{\Large\bf Homework 9 \hfill 21-127 Concepts of Math \hfill due 11.15.2012}\\
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\noindent{\color{blue}\large\bf Name:} % WRITE YOUR NAME HERE
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{\color{blue}\large\bf Andrew ID:} % WRITE YOUR ANDREW ID HERE
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{\color{blue}\large\bf Section:} % WRITE YOUR RECITATION SECTION HERE
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{\color{blue}\large\bf Collaborators:} % LIST ANYONE YOU WORKED WITH HERE
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\begin{problem}[What's The Over/Under? (10 pts)]
Consider the following explanation:
\begin{quote}The number of 6-card hands, as dealt from a standard deck of cards, that have {\it at least one} card from each of the four suits is
\[ \binom{13}{1}\binom{13}{1}\binom{13}{1}\binom{13}{1}\binom{48}{2} \]
because we select one card from each of the four suits and then, from the remaining 48 unused cards, select two more.
\end{quote}
Is this count correct? If you think it is an {\it overcount}, exhibit a specific hand and show how it is counted in two ways. If you think it is an {\it undercount}, exhibit a specific hand and show how it is not counted.
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\begin{problem}[Tame Hat Claim (10 pts)]
Let $X$ be the set of all anagrams of the word MATHEMATICAL.\vspace{0.2cm}\\
Find $|X|$. Be sure to cite the Rule of Product when you use it.\vspace{0.4cm}\\
(For example, the set of anagrams of the word BEE is $\{$\,BEE\,,\,EBE\,,\,EEB\,$\}$)
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\end{problem}
\begin{problem}[We Two Kings (15 pts)]
Let $U$ be the set of all 5-card hands, as dealt from a standard deck of cards, that have exactly two Kings and exactly one Heart.\vspace{0.2cm}\\
Find $|U|$. Be sure to cite the Rules of Sum and Product when you use them.
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\end{problem}
\begin{problem}[String Things (20 pts)]
Let $n\in\N$ (with $n\geq 3$) and let $S$ be the set of all binary strings of length $n$.\vspace{0.2cm}\\
Each of the following expressions is the size of some subset of $S$. For each one, identify such a subset and explain why it works.\vspace{0.4cm}\\
For example, if I were presented with
\[ \binom{n}{3}+\binom{n}{4}+\binom{n}{5}\]
I would say,
\begin{quote}
Let $S_1\subseteq S$ be the set of all strings with either 3 or 4 or 5 positions that are 0s. We can partition this set into the set of strings with exactly $k$ positions that are 0s, for each $k=3,4,5$. In each case, we can find the size of that part by selecting $k$ of the $n$ total positions to be 0s, and fixing the rest to be 1s. By ROS, then, we find that $|S_1|$ is the sum above.
\end{quote}\vspace{0.4cm}
\begin{enumerate}[{\bf\color{blue}(a)}]
\item $\displaystyle{2^{n-2}}$
\item $\displaystyle{2^n-\binom{n}{n}-\binom{n}{n-1}-\binom{n}{n-2}-\binom{n}{n-3}}$
\item $\displaystyle{\binom{n}{2}-\binom{n-1}{1}}$
\item $\displaystyle{\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{k}}$
\end{enumerate}
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\end{problem}
\begin{problem}[Choices, Choices \ldots\; (20 pts)]
Let $n,k,\ell\in\N$ with $n\geq k$ and $k\geq\ell$.\vspace{0.2cm}\\
Prove the following identity by a {\it counting in two ways} argument.
\[ \binom{n}{k}\binom{k}{\ell}=\binom{n}{\ell}\binom{n-\ell}{k-\ell} \]
({\bf Hint:} You might consider using {\it committees}.)
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\end{problem}
\begin{problem}[Huzzah, Pizza! (25 pts)]
Five people---named Eno, Wot, Ethre, Rufo, and Vief---want to order pizzas for a study group session. Each person will order their own personal-sized pizza.\vspace{0.2cm}\\
Fubini's Pizza shop has the following restrictions. There are 5 different types of meat toppings and 8 different types of vegetable toppings available. Any pizza can have at most 2 toppings.\vspace{0.2cm}\\
Eno, Wot, and Ethre are vegetarians and will order veggie pizzas only. (Note: A pizza with 0 toppings has only cheese, and is considered a veggie pizza.)\vspace{0.2cm}\\
Rufo and Vief are devoted meat-eaters and will order meaty pizzas only. (Note: A pizza with, say, pepperoni and mushrooms is considered a meaty pizza.)\vspace{0.2cm}\\
How many possible orders can these people make from Fubini's Pizza shop?\vspace{0.2cm}\vspace{0.2cm}\\
(Bonus: If they spend exactly 10 seconds discussing each possible order, how long it will take them to consider all the possibilities?)
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