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\headers{Math378}{Quiz 10}{Wed. Apr. 8, 2026}
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\noindent
\hfill Name: \underline{\hspace{3.5in}} 

\noindent
\textbf{Directions:} Show all work.  Unless otherwise directed, you may leave your answer in terms of factorials, falling factorials, and binomial/multinomial coefficients.  Any sums or products should be as simple as possible.

\begin{enumerate}

\item\pointsf{2.5} How many $3$-element subsets of $\{a,b,c,d,e,f,g,h\}$ are there?  Give an explicit numerical answer.
\vspace{3cm}

\item\pointsf{2.5} A group of $50$ people must be split into two teams of size $5$ and five teams of size $8$.  How many ways can this be done?
\vspace{3cm}

\item\pointsf{2.5} A random lattice path of length $2n$ is generated by starting from $(0,0)$.  At each step, we increase the $x$-component by $1$ with probability $1/2$ and we increase the $y$-component by $1$ with probability $1/2$.  What is the probability the lattice path ends at $(n,n)$?
\vfill

\item\pointsf{2.5} A standard deck of cards has one card for each of the suit/rank pairs.  The suits are spades, hearts, diamonds, and clubs; the ranks are ace, $2$ through $10$, jack, queen, and king.  What is the probability that a poker hand (i.e. a set of $5$ cards) has at most $2$ cards in each suit?
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\end{enumerate}
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