102 lines
2.5 KiB
TeX
102 lines
2.5 KiB
TeX
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\section{Coins}
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Consider the following problem:
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\say{How many different ways can you make change for \$0.50 \par
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using pennies, nickels, dimes, quarters and half-dollars?}
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\vspace{2mm}
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Most ways of solving this involve awkward brute-force
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approache that don't reveal anything interesting about the problem:
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how can we change our answer if we want to make change for
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\$0.51, or \$1.05, or some other quantity?
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\vspace{2mm}
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We can use generating functions to solve this problem in a general way.
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\definition{}
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Let $p_0, p_1, p_2, ...$ be such that $p_k$ is the number
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of ways to make change for $k$ cents with only pennies.
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Similarly, let...
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\begin{itemize}
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\item $n_k$ be the number of ways to make change for $k$ cents with only nickels;
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\item $d_k$ be the number of ways using only dimes;
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\item $q_k$ be the number of ways using only quarters;
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\item and $h_k$ be the number of ways using only half-dollars.
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\end{itemize}
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\problem{}<pcoins>
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Let $p(x)$ be the generating function that corresponds to $p_n$. \par
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Express $p(x)$ as a rational function.
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\vfill
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\problem{}
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Modify \ref{pcoins} to find expressions for $n(x)$, $d(x)$, $q(x)$, and $h(x)$.
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\vfill
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\pagebreak
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\definition{}
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Now, let $N(x)$ be the generating function for the sequence
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$n_0, n_1, ...$, where $n_k$ is the number of ways to make
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change for $k$ cents using pennies and nickels.
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Similarly, let...
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\begin{itemize}
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\item let $D(x)$ be the generating function for the sequence using pennies, nickels, and dimes;
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\item let $Q(x)$ use pennies, nickels, dimes, and quarters;
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\item and let $H(x)$ use all coins.
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\end{itemize}
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\problem{}
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Express $N(x)$ as a rational function.
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\vfill
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\problem{}
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Using the previous problem, write $D(x)$, then $Q(x)$, then $H(x)$
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as rational functions.
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\vfill
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\problem{}
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Using these generating functions, find recurrence relations for
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the sequences $N_k$, $D_k$, $Q_k$, and $H_k$.
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\hint{
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Your recurrence relation for $N_k$ should refer to the
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previous values of itself and some values of $p_k$.
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Your recurrence for $D_k$ should refer to itself and $N_k$;
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the one for $Q_k$ should refer to itself $D_k$;
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and the one for $H_k$ should refer to itself and $Q_k$.
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}
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\vfill
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\problem{}
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Using these recurrence relations, fill following table
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and solve the original problem.
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\begin{center}
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\begin{tabular}{ c|cccc|cccc|ccc }
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$n$ & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\
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\hline
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$p_k$ &&&&&&&&&& \\
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$N_k$ &&&&&&&&&& \\
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\hline
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$D_k$ &&&&&&&&&& \\
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\hline
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$Q_k$ &&&&&&&&&& \\
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$H_k$ &&&&&&&&&&
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\end{tabular}
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\end{center}
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\vspace{1cm}
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\pagebreak
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