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\lhead{Math 2001 \\ Assignment 9} 
\rhead{Solutions\\ November 29, 2023}

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\begin{enumerate}
\item[1.] This problem involves a deck of $52$ distinct playing cards.
\begin{enumerate}
\item[(a)]
In how many ways can a $13$-card bridge hand be dealt from the deck?
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\item[(b)]
How many different $13$-card bridge hands are there?
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\end{enumerate}  
\end{enumerate}  
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\begin{enumerate}
\item[2.]
\begin{enumerate}
\item[(a)] How many paths are there from the point $(0,0)$
of $\mathbb R^2$
to the point $(10,15)$ of $\mathbb R^2$ if each path 
consists of a sequence of steps of length 1 moving 
in the direction of the positive $x$-axis or the positive $y$-axis?
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\item[(b)] How many paths are there from the point $(0,0,0)$ 
of $\mathbb R^3$
to the point $(10,15,20)$ of $\mathbb R^3$ if each path 
consists of a sequence of steps of length 1 moving 
in the direction of the positive $x$-axis, the positive $y$-axis
or the positive $z$-axis?
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\end{enumerate}
\end{enumerate}
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\begin{enumerate}
\item[3.]  
Let $MC(n,k)$ be the number ``$n$-multichoose-$k$''.
Use a combinatorial argument to show that
$MC(n,0)+MC(n,1)+\cdots+MC(n,k)=MC(n+1,k)$.
\end{enumerate}

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