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\begin{center}\large 
MATH 636, Fall 2014

{\sc Homework 5}

due 5:00{\sc pm} on Thursday, October 30, 2014.
\end{center}

\noindent
{\em Background reading:} {\em Combinatorics: A Guided Tour}, Sections 3.3 and 3.5, and pp. 114--115. \newline  (Ignore any discussion of Exponential Generating Functions.)

\smallskip\noindent
{\bf Follow the posted homework guidelines} when completing this assignment.  \\ There are four questions, each worth four points each.  \\ Please {\bf \color{red}only} consult with your classmates or professor to discuss the problem set. 

\smallskip\noindent
 I would also suggest that you use some computer algebra software (Mathematica, Sage, Maple, Wolfram Alpha, etc.).  When you use the software, {\bf\color{red} let me know} that you are using software, {\bf \color{red}what you input} into the software, and {\bf \color{red}why} the answer you get answers the original question.


\begin{enumerate}
  

\item[\bf 5-1.]  
Consider the scenarios in parts (b), (d), (e), and (f) of Exercise 3.3.2 on page 113.  For each of these {\bf four} scenarios, 
\begin{itemize}
\item[(i)] Find a concise generating function for the situation.
\item[(ii)] Identify the coefficient that you would need to extract to answer the question.
\item[(iii)] Actually extract the coefficient [{\em in parts (d), (e), and (f)}].  
\end{itemize}


\item[\bf 5-2.] 
\begin{enumerate}
\item Algebraically verify the equation
\[\frac{1}{(1-x)(1-x^2)(1-x^3)}=\frac{1}{3(1-x^3)}+\frac{1}{4(1-x^2)}+\frac{1}{4(1-x)^2}+\frac{1}{6(1-x)^3}.\]
\item Use the equation from Part (a) to determine an explicit formula for the number of ways to score $n$ points in a basketball game.
\item How many ways there are to score 100 points in a basketball game?
\end{enumerate}

\item[\bf 5-3.] Exercises 3.5.1(b), 3.5.1(d), and 3.6.1(a)

\item[\bf 5-4.] Use generating functions to prove that for all $n\geq k\geq 1$, \[\sum_{i=0}^k (-1)^i\bbinom{n}{i}\binom{n}{k-i}=0\]

\end{enumerate}




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