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\headers{Math378}{Homework 9}{Due Fri. Mar 27, 2026}

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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 {} [3.1.11] How any numbers in $\{1000,\ldots,9999\}$ are divisible by $5$ and contain the digit $4$?

\item {} You choose a positive integer less than or equal to $10,000$ at random.  What is the probability that your chosen integer:
\begin{enumerate}
	\item {} has all distinct digits?
	\item {} [3.2.4] has exactly one $4$ and one $7$?
	\item {} has at least one $4$ and at least one $7$?  (Hints: Count the complement.  Let $U = \{0000,\ldots,9999\}$, let $A$ be the set of $n\in U$ that have no $4$, and let $B$ be the set of $n\in U$ that have no $7$.  Use that $|A\cup B| = |A| + |B| - |A\cap B|$.)
\end{enumerate}

\item {} A fair coin is tossed $9$ times.  Find the probability that the flips form a palindromic sequence (e.g. HHTTHTTHH).

\item {} [3.2.8] Ten soccer players are standing in a circle and randomly passing a ball.  When a player gets the ball, they can pass the ball to anyone except to the player who just passed them the ball.
\begin{enumerate}
	\item If player $A$ starts the exercise, what is the probability that $A$ receives the third pass?
	\item If player $A$ does not start the exercise, what is the probability that $A$ receives the third pass?
\end{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?


\end{enumerate}
\end{document}
