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\begin{document}
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\large

{\bf MATH-569X \hspace{1cm} HW 1}
\vskip 1em
Due \textbf{Feb 7} by the end of the day.
Just submit it on canvas.
The solution has to be typed (using \LaTeX).


\begin{exercise}
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?
\begin{comment}
TODO

\end{comment}
\end{exercise}


\begin{exercise}
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 two consecutive the same means 11, 22, 00 are the forbidden substrings.)}
\begin{comment}
TODO

\end{comment}
\end{exercise}



\begin{exercise} 
 We are given eight rooks, five of which are red and three of which are blue.
 \par
 (a) In how many ways can the eight rooks be placed on an 8-by-8 chessboard
 so that no two rooks can attack one another?
 \par
 (b) In how many ways can the eight rooks be placed on a 12-by-12 chessboard
 so that no two rooks can attack one another?
\begin{comment}
TODO
\end{comment}
\end{exercise}


\begin{exercise}
There are $n$ sticks lined up in a row, and $k$ of them are to be chosen.\\
(a) How many choices are there?\\
(b) How many choices are there if no two of the chosen sticks can be consecutive?
\begin{comment}
TODO
\end{comment}
\end{exercise}
 


\begin{exercise}%(P. 61, \#11)}{
How many sets of three integers between 1 and 20 are possible if no two consecutive integers are to be in a set?
\begin{comment}
TODO
\end{comment}
\end{exercise}


\begin{exercise}
Combinatorially prove the following binomial identity
$$ \sum_{k=0}^{r} {n+k\choose k} = { n + r + 1 \choose r } .$$
\begin{comment}
TODO
\end{comment}
\end{exercise}



\begin{exercise}%(P. 65, \#38.)}{
 How many integral solutions of
 $$ x_1 + x_2 + x_3 + x_4 = 30 $$
 satisfy $x_1 \geq 2, x_2 \geq 0, x_3 \geq -5,$ and $x_4 \geq 8$?
\\
\emph{(Use substitution to get $ \geq 0$ for all variables.)}
\begin{comment}
TODO
\end{comment}
\end{exercise}





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