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

1. [3.1.11] How any numbers in $\{1000,\dots ,9999\}$ are divisible by $5$ and contain the digit $4$?

2. You choose a positive integer less than or equal to $10,000$ at random. What is the probability that your chosen integer:

   1. has all distinct digits?
   2. [3.2.4] has exactly one $4$ and one $7$?
   3. has at least one $4$ and at least one $7$? (Hints: Count the complement. Let $U=\{0000,\dots ,9999\}$, 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 $|A\cup B|=|A|+|B|-|A\cap B|$.)
3. A fair coin is tossed $9$ times. Find the probability that the flips form a palindromic sequence (e.g. HHTTHTTHH).

4. [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.

   1. If player $A$ starts the exercise, what is the probability that $A$ receives the third pass?
   2. If player $A$ does not start the exercise, what is the probability that $A$ receives the third pass?
5. [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.)
6. [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?