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\headers{Math378}{Homework 9}{Due Fri. Mar 28, 2025}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item {} [4.1.3] A standard deck of playing cards has one card for each suit/rank pair, where the $4$ suits are spades, hearts, diamonds, and clubs and the $13$ ranks are ace, 2 through 10, jack, queen, king.  How many ways are there to order a deck of cards so that all cards with the same suit are next to each other?  (The cards within each suit need not be in order.)

\item {} [4.1.10] You order 10 different books online, 3 of which are for your sister.  The books arrive randomly, one by one.  What is the probability that the books for your sister arrive consecutively?

\item How many circular arrangements of $\{1,\ldots,2n\}$ do not place two even numbers consecutively?


\item {} [4.2.7] You want to write down a $3\times 4$ matrix whose entries are either 0 or 1.  You want the matrix to have 2 or 3 ones (and the rest zeros).  How many such matrices are there?

\item {} [4.2.10] A class of 32 students is asked to split into lab groups.  If there are 10 groups of 3 students and one group of 2 students, how many ways can the groups be formed?

\item {} [4.2.22] In a letter dated Nov 22, 1693, Samuel Pepys asked Isaac Newton a probability question.  He posed three scenarios and wanted to know which one had the greatest chance of success. The scenarios were: throwing six dice in hopes of obtaining at least one 6; throwing 12 dice in hopes of obtaining at least two 6's; and throwing 18 dice in hopes of obtaining at least three 6's.  Calculate the probability of success in each of these cases.



\end{enumerate}
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