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Intermediate/Combinatorics/main.tex
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\section{Getting started} \section{Getting started}
An ordered arrangement of objects is called a \textit{permutation}. An unordered selection\footnotemark{} of objects is called a \textit{combination}. All the following problems involve permutations. An \textbf{ordered} arrangement of objects is called a \textit{permutation}. \par
An \textbf{unordered} selection of objects is called a \textit{combination}\footnotemark{}\hspace{-1ex}. \par
All the following problems involve permutations.
\footnotetext{A \say{combination lock} cares about the order of its digits, so its name is inaccurate. Such an object is actually a \textit{permutation} lock!} \footnotetext{A \say{combination lock} cares about the order of its digits, so its name is inaccurate. Such an object is actually a \textit{permutation} lock!}
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\problem{}<AtoZ> \problem{}<AtoZ>
How many different ways are there to arrange the letters ABCDEFG...XYZ? \par How many different ways are there to arrange the letters ABCDEFG...XYZ? \par
The answer is a \textit{very} big number. You should not fully resolve your answer. \par You don't need to fully evaluate your answer, it is a \textit{very} big number. \par
\hint{When you see a problem that's as big as this one, it's often wise to try and understand a simpler case first. Look at \ref{ABCDE} again, and try to create a general strategy.} \hint{Look at \ref{ABCDE} again, and try to create a general strategy.}
\begin{instructornote} \begin{instructornote}
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First, let's create a function $\npk{n}{k}$, which tells us how many $k$-object permutations we can choose from a group of $n$ objects. First, let's create a function $\npk{n}{k}$, which tells us how many $k$-object permutations we can choose from a group of $n$ objects.
\problem{} \problem{}
What is $\npk{5}{3}$? \\ What is $\npk{5}{3}$? \par
\hint{See \ref{3fromABCDE}} \hint{See \ref{3fromABCDE}}
@ -131,9 +133,9 @@
\path [draw=black, fill=white] (2,0) circle (5pt); \path [draw=black, fill=white] (2,0) circle (5pt);
% "Choose these" bracket % "Choose these" bracket
\draw[shift={(-0.5, -1)}, color=orange, thick] (0pt,0pt) -- (0pt,3pt); \draw[shift={(-0.5, -1)}, color=oblue, thick] (0pt,0pt) -- (0pt,3pt);
\draw[color=orange, thick] (-0.5, -1) -- (2.5, -1) node[below, midway] {Choose $k$ objects}; \draw[color=oblue, thick] (-0.5, -1) -- (2.5, -1) node[below, midway] {Choose $k$ objects};
\draw[shift={(2.5, -1)}, color=orange, thick] (0pt,0pt) -- (0pt,3pt); \draw[shift={(2.5, -1)}, color=oblue, thick] (0pt,0pt) -- (0pt,3pt);
\draw[-] (3, -0.5) -- (3, 0.5); \draw[-] (3, -0.5) -- (3, 0.5);
@ -142,9 +144,9 @@
\path [draw=black, fill=black] (5,0) circle (5pt); \path [draw=black, fill=black] (5,0) circle (5pt);
% "Leave these" bracket % "Leave these" bracket
\draw[shift={(3.5, -1)}, color=orange, thick] (0pt,0pt) -- (0pt,3pt); \draw[shift={(3.5, -1)}, color=oblue, thick] (0pt,0pt) -- (0pt,3pt);
\draw[color=orange, thick] (3.5, -1) -- (5.5, -1) node[below, midway] {Leave the rest}; \draw[color=oblue, thick] (3.5, -1) -- (5.5, -1) node[below, midway] {Leave the rest};
\draw[shift={(5.5, -1)}, color=orange, thick] (0pt,0pt) -- (0pt,3pt); \draw[shift={(5.5, -1)}, color=oblue, thick] (0pt,0pt) -- (0pt,3pt);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
@ -164,10 +166,12 @@
\vfill \vfill
\pagebreak \pagebreak
\section{Combinations}ow m \section{Combinations}
Now, let's count \textit{combinations}. \par Now, let's count \textit{combinations}. \par
Here, we care about \textit{which} items we choose, but not \textit{how} we choose them. We'll make a function $\nck{n}{k}$, or \textit{``n choose k''}. This will tell us how many different ways we can choose $k$ items from a set of $n$. Here, we only care about \textit{which} items we choose---not the order in which we choose them.
We'll make a function $\nck{n}{k}$ (\say{n choose k}), which will tell us
how many different ways we can choose $k$ items from a set of $n$.
\problem{} \problem{}
Find an expression for $\nck{n}{k}$ by modifying your definition of $\npk{n}{k}$. Find an expression for $\nck{n}{k}$ by modifying your definition of $\npk{n}{k}$.
@ -181,13 +185,13 @@
\problem{}<manyballs> \problem{}<manyballs>
Say you have a few coins on the table in font of you: Say you have a few coins on the table in font of you:
\begin{itemize} \begin{itemize}
\item 8 identical 1 kop\footnotemark{} coins \item 8 identical 1-kopek\footnotemark{} coins
\item 3 identical 2 kop coins \item 3 identical 2-kopek coins
\item 6 identical 5 kop coins \item 6 identical 5-kopek coins
\item 4 identical 10 kop coins \item 4 identical 10-kopek coins
\end{itemize} \end{itemize}
How many distinct ways are there to arrange these coins in a row? How many distinct ways are there to arrange these coins in a row?
\footnotetext{Kopek, Russian currency. Comparable to a penny, since 100 kopeks make a ruble.} \footnotetext{Russian currency. Comparable to a penny, since 100 kopeks make a ruble.}
\vfill \vfill
@ -197,8 +201,10 @@
Now, derive the \textit{multinomial coefficient} $\binom{n}{k_1,k_2,...,k_m}$. \par Now, derive the \textit{multinomial coefficient} $\binom{n}{k_1,k_2,...,k_m}$. \par
\vspace{1mm} \vspace{1mm}
The multinomial coefficient tells us how many distinct ways we can choose $n$ objects from a set which has $m$ classes, and where each class $i$ contains $k_i$ identical objects. \par The multinomial coefficient tells us how many distinct ways we can choose $n$ objects from a set which has $m$ classes, and where each class $i$ contains $k_i$ identical objects. \par
\hint{In \ref{manyballs}, $n = 5$ and $(k_1, k_2, k_3, k_4) = (8, 3, 6, 4)$. \\ \hint{
So, the solution to \ref{manyballs} should be given by the multinomial coefficient $\binom{5}{8,3,6,4}$.} In \ref{manyballs}, $n = 5$ and $(k_1, k_2, k_3, k_4) = (8, 3, 6, 4)$. \\
So, the solution to \ref{manyballs} should be given by the multinomial coefficient $\binom{5}{8,3,6,4}$.
}
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\vfill \vfill
\problem{} \problem{}
How many ways can you put 19 identical balls into 6 bins, so that no bin is empty? How many ways can you put 19 identical balls into 6 bins, leaving no bin empty?
\vfill \vfill
@ -243,23 +249,11 @@
How many ways can we split the number 2016 into a sum of positive integers? How many ways can we split the number 2016 into a sum of positive integers?
\vfill \vfill
\pagebreak
\problem{} \problem{}
A staircase must be built up a wall. It will start 4.5 meters away from the wall, which is 1.5 meters tall. The height of each step is exactly 30 centimeters. The width of each step must be an integer multiple of 50 centimeters. In how many ways can the staircase be constructed? A staircase must be built up a wall. It will start 4.5 meters away from the wall, which is 1.5 meters tall. The height of each step is exactly 30 centimeters. The width of each step must be an integer multiple of 50 centimeters. In how many ways can the staircase be constructed?
\vfill \vfill
\pagebreak
\section{Bonus problems}
\problem{}
A toy consists of a ring with 3 red beads and 7 blue beads on it. If two configurations of beads differ only by rotations and reflections, they are considered the same toy. How many different toys are there?
\vfill
\problem{}
At a math circle meeting, 10 students are given 10 problems. Any two students solved a different number of problems, and every problem is solved by the same number of students. Yan solved problems 1 through 5, but did not solve problems 6 through 9. Did he solve problem 10?
\vfill
\end{document} \end{document}