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\headers{Math375}{Homework 3}{Due: Sept 3, 2015}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  \textbf{Show your work}; See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item Poker hands.

\begin{enumerate}
\item A \emph{full house} is a poker hand in which 3 cards have the same rank and the other 2 also have the same rank (e.g. 3 of hearts, 3 of spades, 3 of clubs, 7 of diamonds, 7 of clubs).  How many poker hands are full houses?  What are the odds of being dealt a full house from a freshly shuffled deck?  

Hint: Resist the temptation to look up the answer online.  It is not allowed \emph{and} it defeats the purpose of the question.

\item A \emph{near flush} is a poker hand in which 4 cards belong to a single suit and the remaining card belongs to a different suit.  How many poker hands are near flushes?  What are the odds of being dealt a near flush from a freshly shuffled deck?
\end{enumerate}

\item How many ways are there to arrange the letters of the word `SUSPICIOUS':
\begin{enumerate}
\item with no additional restrictions?
\item if no two S's are consecutive?
\end{enumerate}

\item How many rectangles are created when $n$ horizontal lines intersect $n$ vertical lines?  (Note: a square is a rectangle, so squares are included.)  For example, when $n=3$, there are $9$ rectangles, one of which is highlighted below:
\begin{center}
\begin{tikzpicture}
\foreach \x in {1,2,3}
{{
	\draw (\x, 0.8) -- (\x, 3.2) ;
}}
\foreach \y in {1,2,3}
{{
	\draw (0.8, \y) -- (3.2, \y) ;
}}
\draw[line width=2pt] (1,1) -- (3,1) -- (3,2) -- (1, 2) -- (1,1) ;
\end{tikzpicture}
\end{center}

\item Binomial theorem.
\begin{enumerate}
	\item Find the coefficient of $x^6y^2z^3$ in $(2x-y+3z)^{11}$.
	\item Compute $\sum_{k=0}^n 2^k\binom{n}{k}$.
	\item Compute $\sum_{k=0}^n \frac{1}{k!(n-k)!}$.  Hint: recall the formula for $\binom{n}{k}$.  Relate the given sum to one involving binomial coefficients.
\end{enumerate}
\end{enumerate}
\end{document}
