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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{\myurl/classes/15-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/15-475}
}

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{\large\bf Quiz 7} on March 3, 2015: ``Divide into Cases'', ``Work Backwards'', the Pigeonhole Principle. You have 30 minutes to solve the two problems below. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1} (Larson's 2.5.13).
\begin{itemize}
\item (4 points) A derangement is a permutation $\sigma\in S_n$ such that for every $i$, $\sigma i\neq i$. Let $g_n$ be the number of derangements in $S_n$. Show that
  \[ g_1=0,\qquad g_2=1,\qquad g_n = (n-1)(g_{n-1}+g_{n-2}). \]
  Hint. A derangement interchanges $1$ with some other element, or not.
\item (4 points) Let $f_n$ be the number of permutations in $S_n$ that have exactly one fixed point (namely, exactly one $i$ such that $\sigma i=i$). Show that $|f_n-g_n|=1$.
\end{itemize}

\noindent{\bf Problem 2} (Larson's 2.6.1, 5 points). Given $n+1$ positive integers, none of which exceeds $2n$, show that one of these integers divides another of these integers.

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