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\def\prob{\advance\probno by 1 \bigskip\par\noindent\the\probno . }

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{\usecounter{enumi}\setlength{\rightmargin}{\leftmargin}}}
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\begin{document}

\centerline{A. Miller \hfill M475 \hfill Exam \hfill Spring 2010}

\begin{center}
  Instructions
\end{center}

\underline{Do not hand in this sheet.} Show all work. Write clearly and
carefully.  

\bigskip


% chapt 2 - 15
\prob At a party there are 10 men and 7 women.
\begin{enum}
\item How many ways are there to form 7 couples consisting
of one man and one woman?
\item How many ways are there to form 5 couples consisting
of one man and one woman?
\end{enum}


% chapt 3 - 23
\prob The line segments joining 10 points are arbitrarily colored
red or blue.  Prove that ther must exist three points such that
the three line segments joining them are all red, or four points
such that the six line segments joining them are all blue.
Equivalently, prove that $10\to (3,4)^2$ or $r(3,4)\leq 10$.

(You may assume without proof that
$6\to (3)^2_2$ or equivalently
$r(3,3)\leq 6$.)

% chapt 4 - 8
\prob How many permutations of $\{1,2,3,4,5\}$ have
\begin{enum}
\item exactly 10 inversions?
\item exactly  9 inversions?
\item exactly  2 inversions?
\end{enum}

% chapt 5 - 53
\prob Construct a partition of the subsets of $\{1,2,3,4\}$ into
symmetric chains.


% chapt 5 - 40
\prob What is the coefficient of $x^2y^3z$ in the expansion of
$$(2x-y+4z)^6\;?$$

\newpage
\begin{center}

Answers

\end{center}

1. a. $\comb(10,7)7!$

1. b. $\comb(10,5)\comb(7,5)5!$

2. Fix any vertex $v$.  There are two cases.

\noindent Case 1. There are 4 or more edges from $v$ colored red.

\noindent Case 2. There are 6 or more edges from $v$ colored blue.

3. 1,4,9

4. This is in the book on page 153.

5. -960

\end{document}




