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\centerline{\bf Math 365 -- Monday 3/11/19}

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(We're replacing the old \#26.)

\begin{try}~
\begin{enumerate}[(a)]
\item Consider strings of length 10 consisting of 1's, 2's, and/or 3's. 
\begin{enumerate}[(i)]
\item How many of these are there (with no additional restrictions)?
\item How many of these are there that contain exactly three 1's, two 2's, and five 3's?
\end{enumerate}
\item How many anagrams are there of MISSISSIPPI?
\item Suppose you've got eight varieties of doughnuts to choose from at a doughnuts shop.
\begin{enumerate}[(i)]
\item How many ways can you select 6 doughnuts? 
\item How many ways can you select a dozen (12) doughnuts?
\item How many ways can you select a dozen doughnuts with at least one of each kind?\\
	{[Hint: if there's at least one of each kind, then how many choices are you really making?]}
\end{enumerate}
\item How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a jar contain if it has 20 coins in it?
\item Counting solutions.
\begin{enumerate}[(i)]
\item How many solutions are there to the equation 
$x_1 + x_2 + x_3 = 10,$
where $x_1, x_2$, and $x_3$ are nonnegative integers?
\item How many solutions are there to the equation 
$x_1 + x_2 + x_3 = 10,$
where $x_1, x_2$, and $x_3$ are strictly positive integers?\\
{[Hint: See problem (c)(iii)]}
\item How many solutions are there to the equation 
$x_1 + x_2 + x_3 \leq 10,$
where $x_1, x_2$, and $x_3$ are nonnegative integers?\quad
{[}Hint: Use an extra variable $x_4$ such that $x_1 + x_2 + x_3 + x_4 = 10${]}
\end{enumerate}

\end{enumerate}
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\begin{try}
~
\begin{enumerate}[(a)]
\item List the partitions of 6, both as box diagrams and as sequences. 
\item How many ways are there to distribute 6 identical cookies into 6 identical lunch boxes, possibly leaving some empty?
\item How many ways are there to distribute 6 identical snack bars into 4 identical lunch boxes, possibly leaving some empty?
\item How many ways are there to distribute 4 identical apples into 6 identical lunch boxes, possibly leaving some empty?
\end{enumerate}
\end{try}

\medskip

\begin{try}~
\begin{enumerate}[(a)]
\item Basic counting:
\begin{enumerate}[(a)]
\item[(i)] How many ways are there to distribute 5 distinguishable objects into 3 distinguishable boxes, possibly leaving some empty?

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\item[(ii)] How many ways are there to distribute 5 indistinguishable objects into 3 distinguishable boxes, possibly leaving some empty?

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\item[(iii)] How many ways are there to distribute 5 distinguishable objects into 3 indistinguishable boxes, possibly leaving some empty?

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\item[(iv)] How many ways are there to distribute 5 indistinguishable objects into 3 indistinguishable boxes, possibly leaving some empty?

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\item[(v)] How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes, possibly leaving some empty?

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\item[(vi)] How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes so that each of the boxes contains at least one object?



\end{enumerate}



\item How many ways are there to pack 8 identical DVDs into 5 indistinguishable boxes? How many ways to do this task so that each box contains
at least one DVD?



\item  How many ways are there to distribute 5 balls into 7 boxes if
\begin{enumerate}[(i)]
\item both the balls and boxes are labeled?

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\item the balls are labeled, but the boxes are unlabeled?

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\item the balls are unlabeled, but the boxes are labeled?

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\item both the balls and boxes are unlabeled?
\end{enumerate}



\item Repeat parts (i)--(iv) of part (c), adding the condition that each bucket can have at \emph{most} one ball in it.


\end{enumerate}
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\textbf{\Large Summary of counting techniques:}

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\textbf{\large Placing objects in order (``permutation")}

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\textbf{With replacement:} The number of ways to pick $n$ objects, in order, with possible repetition, from a set of $k$ objects is \fbox{$k^n$}.

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\textbf{With some indistinguishable objects (anagrams):} The number of ways to place $n$ objects consisting of exactly
$$n_1 ~ \text{`$O_1$'s}, \quad n_2 ~  \text{`$O_2$'s}, \quad \dots, \quad \text{ and } n_k ~  \text{`$O_k$'s}$$
(so that $n = n_1 + \cdots + n_k$), in order is \\
\centerline{\fbox{$\displaystyle\binom{n}{n_1} \binom{n-n_1}{n_2}  \cdots \binom{n_r}{n_r} = \frac{n!}{n_1!n_2!n_3!\cdots n_k!}$.}}

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\textbf{\large Placing objects into boxes \\ (``combination'': no order inside the boxes)}

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\textbf{Distinguishable objects, distinguishable boxes:} Distributing $n$ distinguishable objects into $k$ distinguishable boxes is the same as the permutation problems above. If there are no restrictions, then \fbox{$k^n$}. If you restrict to placing exactly $n_i$ objects into box $i$, for $i = 1, 2, \dots, k$ (so that $n = n_1 + \cdots + n_r$), is \\
\centerline{\fbox{$\displaystyle\binom{n}{n_1}\binom{n-n_1}{n_2} \cdots \binom{n_k}{n_k} = \frac{n!}{n_1! n_2! \cdots n_k!}.$}}

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\textbf{Indistinguishable objects, distinguishable boxes:} (\emph{Stars and bars}) The number of ways to distribute $n$ indistinguishable objects into $k$ distinguishable boxes is \\
\centerline{\fbox{$\displaystyle\binom{n+k-1}{n} = \binom{n+k-1}{k-1} =  \frac{(n+k-1)!}{n! (k-1)!}.$}}

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\textbf{Indistinguishable objects, indistinguishable boxes:} (\emph{Integer partitions}) The number of ways to distribute $n$ indistinguishable objects into $k$ indistinguishable boxes is the same of the number of integer partitions of $n$ into at most $k$ parts, \fbox{$p_k(n)$} (there is no closed formula; you just have to count them). 


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\textbf{Distinguishable objects, indistinguishable boxes:} \ The number of ways to distribute $n$ distinguishable objects into $k$ indistinguishable boxes is given by \\
\centerline{\fbox{$\displaystyle\sum_{j=1}^k S(n,j), \quad \text{ where } \quad S(n,j) = \frac{1}{j!} \sum_{i=0}^{j-1} (-1)^i \binom{j}{i} (j-i)^n.$}}
We call the numbers given by $S(n,j)$ the \emph{Serling numbers of the second kind}. 


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{\bf A note on conditions like ``where there's at least one of each kind'', or ``where there's at least one object in each box'', or solving linear equations using \emph{strictly positive} values:} these conditions effectively just decrease the number of choices you're making. 

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{\bf Example:} Choose 10 pieces of fruit from a bowl with indistinguishable apples, oranges, and bananas, making sure to choose at least one of each kind.\\
{\bf Answer:} Effectively, you've already picked out three pieces of fruit: one apple, one orange, and one pear. So you only need to count how many ways you can make the remaining $10-3$ choices, for which you will use stars and bars, with $n = 10 - 3 = 7$ and $k=3$ (the number of kinds of fruit).


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