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\title{General Mathematics and Computational Science I}
\author{Midterm II}
\date{November 13, 2007}
\maketitle

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\def\N{{\mathbb N}}

\begin{enumerate}

\item How many distinct subsets does a finite set with $n$ elements
have?  (The empty set and the set itself are should be included in the
count.) 
\hfill (7)

\item Three dice are tossed. 
\begin{enumerate}
\item What is the probability of obtaining at least one $6$?
\item What is the probability of obtaining at most one $6$?
\end{enumerate}
\hfill (5+5)

\item Use the method of generating functions to find a closed form
expression for the members of the sequence 
\begin{equation*}
  a_{n} = \tfrac52 \, a_{n-1} - a_{n-2}
\end{equation*}
where $a_0=2$ and $a_1=\tfrac52$.
\hfill(10)

% \begin{align*}
%   \phi(x)
%   & = \frac{a_0 + (a_1 - \tfrac52 \, a_0) x}{1-\tfrac52 x + x^2} \\
%   & = \frac{2-\tfrac52 x}{1-\tfrac52 x + x^2} \\
%   & = \frac{1-2x+1-x/2}{(1-x/2)(1-2x)} \\
%   & = \frac1{1-x/2} + \frac1{1-2x} \\
%   & = (1 + x/2 + (x/2)^2 + (x/2)^3 + \dots)
%       + (1 + 2x + (2x)^2 + (2x)^3 + \dots)
% \end{align*}


% \[
%   a_n = \frac1{2^n} + 2^n
% \]


\item Show that, among all right triangles with hypotenuse of length
  $1$, the isosceles triangle has the largest area.
\hfill (8)

\item Show that, for arbitrary numbers $a_1, \dots, a_n$,
\[
  \sum_{k=1}^n a_k 
  \leq n^{\tfrac12} \, \biggl( \sum_{k=1}^n a_k^2 \biggr)^{\tfrac12} \,.
\]
\hfill(7)

\item In class, we computed the probability $P$ that among a group of
  $k$ people at least two are born on the same day of the year.  In
  the derivation, we assumed that that each day has equal probability
  of being the birthday of someone.  In reality, however, there are
  weekly and seasonal variations of the distribution of births.
  Would, under more realistic assumptions on the distribution of
  births, the quantity $P$ increase or decrease?

  \emph{Note:} No credits without explanation.  Full credit for a
  convincing qualitative argument.  Up to five extra credits for a
  complete proof.
\hfill(8+\textit{5})


\end{enumerate}


\end{document}
