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\title{General Mathematics and Computational Science I}
\author{Practice Midterm II -- Not for Credit}
\date{October 27, 2005}
\maketitle

\begin{enumerate}

\item A certain door lock has five buttons.  There is one sequence of
buttons which opens the door.

\begin{enumerate}
\item How many combinations are possible if each button must be
pressed exactly once?

\item Assume that there exists, in addition, sequences where one or
more pairs of buttons must be pressed simultaneously.  How many
combinations are possible now?

\end{enumerate}

\item Show that 
\begin{equation*}
  \sum_{j=0}^k \binom{m}{j} \binom{n-m}{k-j} = \binom{n}{k} 
\end{equation*}
for integers $0 \leq k \leq m \leq n$.

(Use the generating function for the binomial coefficients.)

\item Solve the ``hat check problem'' (see separate handout).

\item Solve the recurrence relation
\begin{equation*}
  a_n - 5 \, a_{n-1} + 6 \, a_{n-2} = 0
\end{equation*}
with $a_0=2$ and $a_1=5$, using the method of generating functions. 

\item Ivanov, p.~46, Problem 1. (Try a selection of subproblems.)

\item Use the $n$-term arithmetic-geometric-mean inequality to derive
Bernoulli's inequality
\begin{equation*}
  (1+x)^n \geq 1 + n \, x
\end{equation*}
for $x \geq 0$ and $n$ a natural number.



  

\end{enumerate}
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