\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{12b, 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 B
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\begin{problem}
A deck contains $n$ red cards (numbered $1, \ldots, n$) and $2n$ black
cards (numbered $1, \ldots, 2n$).

\bparts

\ppart In how many ways can we select $n$ cards, if exactly $r$ of
them must be red and the remaining $n - r$ must be black?

\solution[\vspace{1in}]{
\[
\binom{n}{r} \cdot \binom{2n}{n - r}
\]
}

\ppart Suppose we select $n$ cards from the deck.  The number of red
cards that we select, $r$, must be between 0 and $n$.  Use the Sum
Rule and your previous answer to determine the total number of ways to
select $n$ cards.  (You may leave your answer as a summation.)

\solution[\vspace{0.75in}]{
\[
\sum_{r=0}{n} \binom{n}{r} \cdot \binom{2n}{n - r}
\]
}

\eparts

\end{problem}

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\begin{problem}
Grant Wang, famed 6.042 TA, has decided to try out for the US olympic
boxing team.  To prepare, 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.  In how many
different ways can the team be selected?

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

\end{problem}

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%END VERSION B

\input{cp12f-common}

\end{document}
