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\headers{Math378}{Quiz 8}{Wed. Mar 25, 2026}
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\noindent
\hfill Name: \underline{\hspace{3.5in}} 

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

\begin{enumerate}

\item\pointsf{2.5} How many ways are there to make a list $(x_1,\ldots,x_5)$ where each $x_i$ is a digit in $\{0,\ldots,9\}$ and we have $4$ even digits and one odd digit?  For example, $(0,0,1,8,6)$ and $(2,2,4,6,3)$ count but $(0,1,2,3,4)$ and $(0,0,0,0,0)$ do not.
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\item\pointsf{2.5} How many integers in $\{1,\ldots,1000\}$ have no repeated consecutive digits?
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\item\pointsf{2.5} Count the functions $f\st \{1,\ldots,9\} \to \{1,2,3,4\}$ such that $f(1) = f(t)$ for some $t\in\{2,\ldots,9\}$.
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\item\pointsf{2.5} How many integers in $\{1000,\ldots,5000\}$ are even and have all distinct digits?
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\end{enumerate}
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