\documentclass[11pt]{article}

\usepackage{times,mathptm}
\usepackage{pifont}
\usepackage{exscale}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{epsfig}

\textwidth 6.5in
\textheight 9in
\oddsidemargin -0.0in
\topmargin -0.0in

\parindent 0pt     % How much the first word of a paragraph is indented. 
\parskip 0pt       % How much extra space to leave between paragraphs.

\begin{document}

\begin{center}             % If you only centering 1 line use \centerline{}
\begin{LARGE}
{\bf CSci 243 Homework 8}
\end{LARGE}
\vskip 0.25cm      % vertical skip (0.25 cm)

**My name**
\end{center}

\begin{enumerate}

\item A DNA sequence is a string over the alphabet of $\{A, C, G, T\}$, 
which are called bases. How many 4-element DNA sequences
\begin{enumerate}
\item (2 points) do not contain the base $T$?
\item (2 points) do not contain the sequence $ACG$?
\item (2 points) contain all four bases, and $AC$ is in the sequence? 
\item (2 points) begin with $A$ or ends with $T$?
\item (4 points) contain exactly three of the four bases $A, T, C,$ and $G$?
\end{enumerate}

\item How many functions are there from the set $\{a_1, a_2, ..., a_n\}$, to the set $\{b_1, b_2, ..., b_m\}$, where $n$ and $m$ are positive integers,
\begin{enumerate}
    \item (2 points) that are one-to-one?
    \item (2 points) that $f(a_1) = b_1$ and $f(a_n) = b_1$?
    \item (4 points) that $f(a_n) > f(a_1)$?
\end{enumerate}

\item (4 points each) A bowl contains 10 yellow marbles and 10 green marbles.
You select marbles at random without looking at them.
\begin{enumerate}
\item How many marbles must you select to be sure of having at least three marbles of the same color? 
\item How many marbles must you select to be sure of having at least three green marbles? 
\end{enumerate}

\item (8 points) How many ordered pairs of integers $(a,b)$ are needed to guarantee that
there are two ordered pairs $(a_1,b_1)$ and $(a_2,b_2)$ such that $a_1\bmod 5=a_2\bmod 5$ and
$b_1\bmod 5=b_2\bmod 5$? 

\item (8 points) How many numbers must be selected from the set $\{1,3,5,7,9,11,13,15\}$ to guarantee that at least one pair of these numbers add up to 16?

\item (6 points) A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers. 



\item (\textbf{Bonus Question} 2 points\footnote{The points for bonus questions are added directly to the final grade.}) How many 8-bit strings contain either three consecutive 0's or four consecutive 1's? Explain your solution. 
 

\end{enumerate}

\end{document}
