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\headers{Math378}{Quiz 12}{Wed. Apr 22, 2026}
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\noindent
\hfill Name: \underline{\hspace{3.5in}} 

\noindent
\textbf{Directions:} Show all work.  

\begin{enumerate}

\item Let $U=[n]^2=\{(x,y)\st x,y\in\{1,\ldots,n\}\}$.  
\begin{enumerate}
    \item\points{4} Use $U$ to give a combinatorial proof that $n^2 = 2\binom{n}{2} + n$.
    \vfill
    \item\points{3} Use part (a) and an identity from HW12 to give a formula for $\sum_{k=1}^n k^2$.
    \vspace{4cm}
\end{enumerate}

\item\points{3} Let $n$ be a positive integer.  Let $A$ be the set of all triples $\{x,y,z\} \in \binom{[n]}{3}$ such that $x<y<z$ and $z-y = y-x$.  Let $B$ be the set of all pairs $\{a,b\}\in \binom{[n]}{2}$ such that $a<b$ and $b-a$ is even.  Give a bijection to show that $|A| = |B|$.
\vfill

\end{enumerate}
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