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\usepackage{amssymb,latexsym,epsfig,anysize}

\def\arrow{\to}

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%\input macros

\def\fillin{\par\noindent
answer:$\underline{\phantom{AAAAAAAAAAAAAAAA}}$
\par\noindent}
\def\rvec#1{\stackrel{\rightarrow}{#1}}
\def\lvec#1{\stackrel{\leftarrow}{#1}}

\newcounter{xprobno}
\newcommand{\xprob}[1]{\addtocounter{xprobno}{1}
   \par\vspace{.15in}\noindent\thexprobno . \marginpar{(#1 points)} }

\begin{document}

\begin{flushright}
  Name $\underline{\phantom{AAAAAAAAAAAAAAAAAAAAAAAA}}$ \hfill  A. Miller\\
  Score  $\underline{\phantom{AAAAAA}}$ \hfill M475 Exam 1 \\
  Grade $\underline{\phantom{AAAAAA}}$ \hfill Oct 7, 1993
\end{flushright}

{\bf Put answers to problems 1-6 on your own paper.}

%%%%%%%%%%%%%%%%%%%%%%%%%% chapter 2

\xprob{12} A sequence of twelve positive integers sums to 103.  Show that
there must be two consecutive elements of the sequence which sum
to at least 18. (Explain your answer.)


\xprob{11} How many three letter words over $\{A,B,C,\ldots, Z\}$
do we need to choose to be sure that two of them agree in at
least one place?  (Note: letters may be repeated, e.g., BBA, XXX are
allowed. BAB and CAD agree in the second place. ABC and
BCA do not agree in any place.) (Explain your answer.)


\xprob{11} Suppose that each edge of the complete graph
on 10 vertices, $K_{10}$, is colored either red or blue.
Show that there must be either three vertices
such that all of the  edges connecting them are colored red (a red
triangle), or four vertices such that all of the
edges connecting them are colored blue (a blue square).
(You may assume without proof that $K_6\arrow (K_3,K_3)$.)

%%%%%%%%%%%%%%%%%%%%%%% chapter 3


\xprob{11} How many paths are there from $(0,0,0)$ to $(3,4,5)$ if
each move is one of the following:
\begin{itemize}
  \item $(x,y,z)\arrow (x+1,y,z)$, or
  \item $(x,y,z)\arrow (x,y+1,z)$, or
  \item $(x,y,z)\arrow (x,y,z+1)$.
\end{itemize}
(Explain your answer.)

\xprob{11} How many 8 letter words can be made by permuting
the 8 letters:
$$a,b,c,d,e,f,f,f$$
if no two $f'$s are to be next to each other?  (Explain your answer.)

\xprob{12} How many combinations of six taken from the numbers
$\{1,2,3,\ldots., 45\}$ are there such that at least two of the six
numbers are consecutive?  (Explain your answer.)

%%%%%%%%%%%%% chapter 4

\newpage

\begin{center}
  In the following problems do {\bf not} explain your answer.\\
  Put your answers in the space provided.\\



\end{center}


\xprob{5} In the lexicographical listing of all permutations
of $\{1,2,\ldots,9\}$ what permutation comes just after
$\{2,1,3, 7,9,8,6,5,4\}$? \fillin


\xprob{6} What are the mobile integers in
$\lvec{7}\;\rvec{6}\;\lvec{5}\;\rvec{3}\;
\lvec{1}\;\lvec{2}\;\lvec{4}\;\rvec{8}\;\rvec{9}$?
\fillin
What is the next permutation after it (including the arrows)?
\fillin

\xprob{6} The sequence $6,3,2,4,0,2,1,0$ is an inversion sequence
for a permutation of $\{1,2,\ldots,8\}$.  What is this permutation?
\fillin  What is the inversion sequence for the
permutation $2,4,7,1,8,3,6,5$?\fillin

\xprob{5} The subsets of $\{x_5,x_4,x_3,x_2,x_1,x_0\}$ correspond to
the numbers $0,1,\ldots, 63$ using the base 2 arithmetic scheme.
What subset corresponds to the number 59? \fillin

\xprob{5} In the lexicographical listing of all 7-combinations
of the set $\{1,2,\ldots,15\}$ what 7-combination comes just
after $\{5,9,10,12,13,14,15\}$?
\fillin

\xprob{5} List all subsets of the set $\{x,y,z\}$ in such a
way that if $A_1,A_2,\ldots,$ is your listing, then
for every $i$ the set $A_{i+1}$ differs from the set $A_i$ by
either adding one element or taking one element away.

\bigskip

\par\noindent answer:



\end{document}