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\begin{document}

\maketitle

\begin{center}
{{\Large\bf MODERN ALGEBRA II}\\
\medskip

{\Large\bf HOMEWORK ASSIGNMENT IX}\\
\medskip
{\bf (Turn in underlined problems.)}}
\end{center}

\vspace{.15in}

\noindent
{\bf 
\begin{tabbing}
\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

\bigskip
\noindent
 \> \underline{2.76}, 2.77
\end{tabbing}

}
\vspace{.15in}

\noindent
\underline{\large\bf ADDITIONAL PROBLEMS}
\vspace{.15in}

{\bf 

\underline{1}. How many essentially different bracelets can be made
from nine pearls which are either white or black?

\bigskip

%How many essentially different ways are there to put
%the numbers 1--6 on the six sides of a playing die?
%How many ways are there if opposite sides must sum to seven?

%\bigskip

\underline{2}. 
How many essentially different ways are there to put
the numbers 1--12 on the twelve sides of a dodecahedron?
How many ways are there if opposite sides must sum to 13?

\bigskip

\underline{3}. You have piles of identical car keys,
identical skeleton keys, and identical skate keys.
How many essentially different ways are there to 
create a key ring of 19 keys from these keys?

\bigskip

\underline{4}. How many essentially different ways are there to 
color the vertices of the following
graph using only colors from the set
$\{\mbox{\rm red, orange, yellow, green, blue}\}$? 
%(Hint: The group of edge-preserving permutations of the vertices
%is isomorphic to $D_4$. One such isomorphism is determined
%by taking $r=(1\;2\;3\;4)(5\;6)$ and 
%$f=(1\;2)(3\;4)(5\;6)$.)


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\bigskip

5. How many essentially different ways are there to 
color the edges of a tetrahedron using only colors
from a set of $q$ colors?

\bigskip

For Graduate Credit: 

I. How many essentially different
ways are there to color the faces of cube red,
green or blue if each color must be used at least once?

\end{document}

