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\documentclass[12pt]{article}

\oddsidemargin 5pt \evensidemargin 5pt \marginparwidth 68pt
\marginparsep 10pt \topmargin 10pt \headheight 10pt
\headsep 15pt \footskip 35pt
\textheight = 625pt  \textwidth 440pt \columnsep 10pt \columnseprule 5pt

\pagestyle{myheadings}
\def\la{\langle}  \def\ra{\rangle}

\def\ansbox{\fbox{\rule[-.25in]{0in}{.5in} answer}}

\def\header{{\hfill Final \hfill A. Miller \hfill Spring  99
                                       \hfill Math 210\hfill}}
\markboth\header\header
\newcount\probno
\newcount\total
\newwrite\examTBLFile

\probno=0
\def\prob#1{\advance\probno by 1 \newpage  % \par
    \noindent\the\probno . (#1 pts) \advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}

\def\probx#1{\advance\probno by 1 % \newpage  % \par
    \noindent\the\probno . \advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}
\def\finishtbl{
    \immediate\write\examTBLFile{Total \string& \the\total
    \string& \qquad \string\\ \string\hline}
     }

\newwrite\ans
\immediate\openout\ans=answers
\outer\def\answer{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\putanswer{\par\medbreak
  \immediate\write\ans{}
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copyans}
\def\setupcopy{\def\do##1{\catcode`##1=12}\dospecials
     \catcode`\|=12\obeylines}
{\obeylines \gdef\copyans#1
  {\def\next{#1}%
  \ifx\next\empty\let\next=\endgroup%
  \else\immediate\write\ans{\next}\let\next=\copyans\fi\next}}
\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

\begin{document}

\begin{center}
No calculators.  \\
No note or index cards.   \\
Show all work.      \\
% Put your answer in the answer box. \\
\end{center}

\par\vskip .25in

\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\bigskip \bigskip \bigskip

% end file with \finishtbl
\bigskip
\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
  \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

\def\medium{20}
\def\high{25}
\def\ch(#1,#2){\left(\begin{array}{c} #1 \\ #2 \end{array}\right)}

\setcounter{page}{0}

\prob{\medium}
A committee of 5 persons is to be formed from a group of 120 students.
80 of these students are Freshmen and 40 of them are Sophomores.
How many possible committees are there in which there are more
Sophomores than Freshmen but at least one Freshman?
\answer $\ch(40,4)\times\ch(80,1)+\ch(40,3)\times\ch(80,2)$

\prob{\high}
In a random group of $5$ people what is the probability that at
least two have the same birthday?
\answer $1-{{365\times 364\times 363\times 362\times 361}\over {365^5}}$

\prob{\high}
Every day Professor NoodleBrain tosses a deck of cards one at
a time at his hat trying to land the cards inside.
After many years of practice the Professor has found that
on average he lands about 13 of the 52 cards in his hat.
\par\bigskip (a) What is the probability, $p$,  that after tossing
all 52 cards that at least 3 of the 4 aces lands in the hat?
\par\bigskip
\par (b) If he repeats this for 10 days in a row, what is the
probability that on exactly 4 of the 10 days he gets at
least 3 of the 4 aces in the hat?  (You may express your answer in
terms of $p$ where $p$ is from part (a).)
\answer (a) $p=({1\over 4})^4+4\times({1\over 4})^3\times({3\over 4})$
\par (b) $\ch(10,4)p^4(1-p)^6$

\prob{\medium}
The axioms for a probability, $Pr(E)$ for
$E\subseteq S$, are the following:
\par (1) $0\leq Pr(E)\leq 1$ for each $E\subseteq S$,
\par (2) $Pr(S)=1$, and
\par (3) if $E_1$ and $E_2$ are disjoint subsets of $S$, then
$Pr(E_1\cup E_2)=Pr(E_1)+Pr(E_2)$.
\par\bigskip\noindent Using these axioms deduce the formula
$$Pr(A\cup B)=Pr(A)+Pr(B)-Pr(A\cap B)$$
where $A$ and $B$ are any subsets of $S$.
\answer See Exam 2 answers.


\prob{\high} Consider the set of solutions of
the system of equations
$$\begin{array}{rcrcrcr}
  x  &+& 2y &+&   z  &=&  6\\
 2x  &+&  y &+&   z  &=&  4 \\
 ax  &+& 3y &+&  2z  &=&  b \\
\end{array}$$
Where $a$ and $b$ are constants.
\par\bigskip
(a) Give an example of an $a$ and $b$ such that this system
has infinitely many solutions.
\par\bigskip $a=$ \hspace{2in}  $b=$
\par\bigskip (b) Give an example of $a$ and $b$ such that this system
has no solution.
\par\bigskip $a=$ \hspace{2in}  $b=$
\answer (a) $a=3\;\;\;b=10$
\par (b) $a=3\;\;\;b$ anything except 10.

\prob{\medium} Find a 3 by 3 matrix $B$ so that $ABA=I$ where $I$ is
the 3 by 3 identity matrix and
\par\bigskip
$A= \left[\begin{array}{rrr}
     1   &   -1    &    1     \\
    -1   &    0    &   -1     \\
     1   &   -1    &   -1     \\
\end{array}\right] \;\;\;\;
I= \left[\begin{array}{rrr}
     1   &   0  &    0     \\
     0   &   1  &    0     \\
     0   &   0  &    1     \\
\end{array}\right] \;\;\;\;
B=$
\answer $B=A^{-1}\times A^{-1}$
$$A^{-1}= \left[\begin{array}{rrr}
       -{1\over 2}  &   -1    &    {1\over 2}  \\
       -1   &    -1    &   0     \\
        {1\over 2}   &   0    &   -{1\over 2}     \\
\end{array}\right]$$
$$B= \left[\begin{array}{rrr}
       1{1\over 2}  &   1 {1\over 2}    &   - {1\over 2}  \\
       1{1\over 2}   &    2    &   -{1\over 2}     \\
      -{1\over 2}   &  -{1\over 2}    &   {1\over 2}     \\
\end{array}\right]$$

\prob{\high}
On January 1, 1998 you deposited \$3000 in a savings account
which pays 4\% compounded quarterly.  On July 1, 1998 you
withdrew \$1000. On July 1,1999 how much will you have in
this account?
\answer $(3000(1.01)^2-1000)(1.01)^4$

\prob{\medium}
On January 1, 2000 you buy a home for \$150,000.  After paying
a down payment of \$30,000 the remaining amount is borrowed
from a bank at 7 \% compounded monthly for 30 years.  The
first monthly payment is paid on January 1, 2000.   What is the
amount of each monthly payment?
\answer ${120000}\over y$ where $y={{1-x^{360}}\over{1-x}}$
and $x={1\over{1+i}}$ and $i={.07\over 12}$.

\prob{\medium}
You wish to save money for your child's college education.
You contribute $2000$ each year to a savings account
which pays 8 \%  annual interest.
Your first contribution is made on January 1, 2000, the
second on January 1, 2001, and so on,  making the same contribution
each January.
You make your last contribution on January 1, 2009.
At that time, how much will your child have in this
savings account?
\answer $2000({{1-x^{10}}\over{1-x}})$ where $x=1.08$.

\finishtbl
\printanswers

\end{document}