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{\bf Worksheet \#16; date: 10/22/2018}
{\bf MATH 55 Discrete Mathematics}
\begin{enumerate}
\item What are the numbers of way to put $42$ balls into $5$ bins if
\begin{enumerate}
\item the balls are identical but the bins are not?
\item the bins are identical but the balls are not?
\item the balls and the bins are all identical?
\item the balls and the bins are all distinct from each other?
\end{enumerate}
\item {\em (Rosen 6.5.9e, f)} A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernickel bagels. How many ways are there to choose
\begin{enumerate}
\item a dozen bagels with at least one of each kind?
\item a dozen bagels with at least three egg bagels and no more than two salty bagels?
\end{enumerate}
\item {\em (Rosen 6.5.16b, c)} How many solutions are there to the equation
\[
x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 29,
\]
where $x_i$, $i = 1, 2, 3, 4, 5, 6$, is a nonnegative integer such that
\begin{enumerate}
\item $x_1 \le 5$?
\item $x_1 > 8$ and $x_2 > 8$?
\end{enumerate}
\item {\em (Rosen 6.5.39)} How many ways are there to travel in $xyz$-space from the origin $(0, 0, 0)$ to the point $(4, 3, 5)$ by taking steps one unit in the positive $x$ directio, one unit in the positive $y$ direction, or one unit in the positive $z$ direction? (Moving in the negative $x$-, $y$- or $z$-direction is prohibited so that no backtracking is allowed.)
\end{enumerate}
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