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\begin{document}

\tutprobn{9}{4/9/99}{April 7, 1999}

\section*{Team Problems}
\begin{problems}

\problem
How many poker hands are there with no pairs?

\problem

\bparts

\ppart

In lecture we demonstrated how to ``guess'' the fifth card in a poker hand
when a collaborator reveals the other four cards.  Describe a method for
guessing two hidden cards in a hand of nine cards, when your collaborator
reveals the other seven cards.

\ppart Show that there is no procedure for ``guessing'' a hidden card
among four when your collaborator reveals three. \hint Compare the number
of hands to the number of revealed three-card sequences.

\eparts

\problem
\bparts

\ppart What are necessary and sufficient conditions on the integer values
of $x_1,y_1,x_2,y_2$ for there to be a count-path---of any kind, not just
one corresponding to a well-matched string---from $(x_1,y_1)$ to
$(x_2,y_2)$?

\ppart How many count-paths of any kind are there from $(x_1,y_1)$ to
$(x_2,y_2)$?  Express the answer as a binomial coefficient.

\ppart For $y_1,y_2 > 0$, show that the number count-paths from
$(x_1,y_1)$ to $(x_2,y_2)$ \emph{which touch or cross the x-axis} is equal
to the number of count-paths of any kind from $(x_1,-y_1)$ to $(x_2,y_2)$.

\ppart Conclude that the number of well-matched strings of $2n$ parentheses
is
\[\frac{1}{n} \binom{2n-2}{n-1}.\]
This number is called the $n-1$st \emph{Catalan number}.

\eparts

\end{problems}

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