\documentclass[12pt]{article}  % This is just a beginning - saying what type of file are we using (everything after % is a comment!)
\usepackage{fullpage}             % Uses all space on the paper 
\usepackage{tikz}                     % For drawing pictures

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% There is also latex in the computer lab in the fourth floor in Carver Hall.
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% You are welcome to use any other template for your hw. This is just to
% Save you some suffering. 
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%  The following is creating a macro question - it does automatic numbering
\newcounter{excounter}
\setcounter{excounter}{1}
\newcommand\question[2]{\vskip 1em  \noindent\textbf{\arabic{excounter}\addtocounter{excounter}{1}:} \emph{#1} \noindent#2}


\begin{document} %% The document starts after this

\large{\bf MATH-304 \hspace{1cm} HW 1  \hfill   Chuck Norris }     %% Change Chuck Norris to your name!!!!!
\vskip 1em
Due \textbf{Sep 3} before class. Just bring it before the class and it will be collected there.
The solution has to be typed (using \LaTeX).

\question{}{
How many ways are there to pick a man and a woman who are not husband and wife from a group of $n$ married couples?
}

\textbf{Solution:}
% Write solution of the problem here!!
% Special characters are:  { } # $ _  ~ \  
% If you need to write them, you can use  \{  to write {
This is just text. Bla bla bla. $x+y$ look nicer - they are mathematics like. More examples
\[
\frac{34!}{x^2}   \cdot    y_{4+4}   \cdot   34^{x + y^2 } 
\]
It creates a fraction and also some stuff around like exponents and subscripts. 


\question{}{
How many nonempty words can be formed from three $A$s and five $B$'s?\\
\emph{(not all letters must be used, any sequence of letters counts as a word)}
}

\textbf{Solution:}
% Write solution of the problem here!!



\question{}{
How many ternary (0,1,2) sequences of length 10 are there without any two consecutive digits \textbf{being} the same? \\
\emph{(Ternary means using digits 0,1,2. Similarly, binary would mean just digits 0,1. No consecutive the same means 11, 22, 00 are the forbidden substrings.)}
}

\textbf{Solution:}
% Write solution of the problem here!!



\question{}{
How many different outcomes are possible when a pair of dice, one red and one white are rolled two consecutive times?\\
Consider that one roll consists of outcome that is formed by the PAIR of what is on white and red and
count the number of outcomes if\\
A) it is possible to distinguish which roll was first and which was second.\\
B) it is not possible to distinguish first and second roll.
\emph{(Consider that first and second roll are distinguishable as well as the case where
they are not. Note that one roll is, which is red and white dices is equivalent to a roll of a dice with 36 sides.)}
}

\textbf{Solution:}
% Write solution of the problem here!!



\question{}{
Construct a perfect cover of an 8-by-8 chessboard with dominoes ($1\times 2$) having no fault-line. \\
\emph{(Fault line is cutting the board but not any domino. See Page 7 in the book and Figure 1.5.)}
}

\textbf{Solution:}
%  Write solution of the problem here!!
%  You may want to draw a picture. Drawing pictures is a pain  :-) Feel free to draw by hand. 
% If you need to create some vertical space, you can use command:    \vskip 4cm   which makes space of 4 cm
% In case you want to include a picture, you can use command \includegraphics{picture.png}
% Or for brave people, there is tikz.



\question{}{
Show that there is no magic cube of dimension 2.\\
\emph{(It means cube $2\times2\times2$ filled with numbers $1\ldots8$.)}
}

\textbf{Solution:}
%% Write solution of the problem here!!


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