\documentclass{article}
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\usepackage{amsmath,amssymb,amsthm}
\usepackage{xcolor}
\newcommand{\XXX}{{\color{red} \textsf{\textbf{XXX}}}}
\begin{document}
% To run this file, you'll need to have LaTeX installed. All of the
% lab computers have it. You can download a complete LaTeX
% distribution from
% https://www.tug.org/texlive/acquire-netinstall.html.
% Once you have LaTeX, you can either build your PDF on the
% command-line by running 'pdflatex hwXX.tex' to generate hwXX.pdf, or
% you can use an editor like LyX, TeXShop, or ShareLaTeX which
% automates the building of your PDF.
% Please print out your solution double-sided (a/k/a duplex) and bring
% it to class on the Wednesday it's due.
\noindent
% fill in the XXXs below
{\Large CS055 HW08 \qquad Name: \XXX \qquad CAS ID: \XXX} \\[.5em]
% I encourage you to collaborate, but please list any other students
% you talked to about the homework. If you worked alone, please just
% remove the XXXs.
Collaborators: \XXX
% Okay! Solve the problems below. please don't delete the problem
% statement or the ``enumerate'' bracketing which provides the
% numbering.
\begin{enumerate}
\item Suppose you're eating from a bowl of olives containing 8
castelvetrano olives and 8 kalamata olives. You take olives at
random, not seeing which you eat until you put it into your
mouth. (Watch out for pits!)
\begin{enumerate}
\item How many olives do you have to eat to ensure you've eaten at
least three of the same kind of olive?
\textbf{Answer:} \XXX
\item How many olives do you need to eat to ensure you've eaten at
least three castelvetrano olives?
\textbf{Answer:} \XXX
\end{enumerate}
\item
%
\newcommand{\C}[2]{\ensuremath{\left( \begin{array}{@{}c@{}} {#1} \\ {#2} \end{array} \right)}}
%
Find a closed form solution for $\sum_{k=0}^{n} \C{n}{k} 2^n$. Show enough work to explain your answer, but no need for proof.
\textbf{Answer:} \XXX
\item Prove the binomial theorem using induction.
\textbf{Theorem:} $(x+y)^n = \sum_{k=0}^n \C{n}{k} x^{n-k}y^k$.
\textbf{Proof:} \XXX
\item The following questions concern a lottery game where you must
choose six distinct numbers between 1 and 47. Order doesn't matter
and the winning numbers are chosen at random.
\begin{enumerate}
\item What is the probability of choosing all six winning numbers?
\textbf{Answer:} \XXX
\item What is the probability of choosing no winning numbers?
\textbf{Answer:} \XXX
\item What is the probability of choosing exactly one winning number?
\textbf{Answer:} \XXX
\item \label{formula} Write down a function $f(n)$ that gives the
probability of having exactly $n$ winning numbers, for $0 \le n
\le 6$.
To check your work, you can verify that $\sum_{n=0}^6 f(n) =
1$. (No need to write down that computation, though! A computer
will make this process faster, but there's an even faster way...)
\textbf{Answer:} \XXX
\end{enumerate}
\end{enumerate}
\end{document}