\documentclass[11pt,twoside]{article}

%\usepackage{course}% !! use the macros at home
\usepackage{latex-macros/course}% !! use the macros

\newcommand{\rem}{\textup{ rem }}

\begin{document} 
\inclassproblems{12a, Fri.}

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%  Note:  There are two versions of this handout.  The plan is to get
%  half the students to count the elements in a set one way and the
%  other half to count another way.  We conclude that both answers are
%  correct, which illustrates the notion of combinatorial proof.

%BEGIN VERSION A
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{problem}
Grant Wang, famed 6.042 TA, has decided to try out for the US olympic
boxing team.  After all, he's watched all of the \textit{Rocky} movies
and spent hours in front of a mirror sneering, ``Yo, you wanna piece
a' \textit{me}?!''.

Grant figures that $n$ people (including himself) are competing for
spots on the team and only $k$ will be selected.  There are only two
possibilities:

\bparts

\ppart Grant is selected for the team, and his $k - 1$ teammates are selected
from among the other $n - 1$ competitors.  In how many different ways
can this happen?

\solution[\vspace{0.5in}]{
\[
\binom{n-1}{k-1}
\]
}

\ppart Grant is not selected for the team, and all $k$ team members
are selected from among the other $n - 1$ competitors.  In how many
different ways can this happen?

\solution[\vspace{0.5in}]{
\[
\binom{n - 1}{k}
\]
}

\ppart Use the sum rule and your answers to the two previous problem
parts to determine all possible ways to select the US olympic boxing
team.

\solution[\vspace{0.5in}]{
\[
\binom{n-1}{k-1} + \binom{n - 1}{k}
\]
}

\eparts

\end{problem}

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\begin{problem}
A deck contains $n$ red cards (numbered $1, \ldots, n$) and $2n$ black
cards (numbered $1, \ldots, 2n$).  In how many different ways can $n$
different cards be selected from this deck?

\solution{
\[
\binom{3n}{n}
\]
}
\end{problem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%END VERSION A

\input{cp12f-common}

\end{document}
