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1. Arrangements of six letters ●○○○○ 10 ptsHow many permutations can be made from the set $\{A, E, I, O, U, Y\}$ using all six letters?
2. Three-letter words from LOVE ●○○○○ 10 ptsHow many 3-letter arrangements (words need not be meaningful) can we make using the letters of the word LOVE, each letter used at most once?
3. Ace of spades before face cards ●●○○○ 20 ptsCarson flips over the cards of a standard 52-card deck one at a time. Find the probability that he flips over the ace of spades before any face card (jack, queen or king). Round to two decimal places.
4. Aces before the joker ●●○○○ 20 ptsA standard 52-card deck plus one joker (53 cards in total) is shuffled into a uniformly random order and turned over one card at a time. What is the probability that all 4 aces are turned over before the joker? Give your answer as a decimal.
5. Strings with distinctness constraints ●●○○○ 20 ptsAlec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: - the first three letters are pairwise distinct, and so are the last three letters; - the first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construct the string?
6. Permutations with order dividing twelve ●●○○○ 20 ptsLet $A=\{1,2,\dots,6\}$ and let $f: A \to A$ be a one-to-one mapping. Define $f^{(2)}(x)=f(f(x))$ and $f^{(n+1)}(x)=f(f^{(n)}(x))$ for $n \ge 1$. Find the number of mappings $f$ such that $f^{(12)}(x)=x$ for every $x \in A$.
7. Aces of spades and clubs adjacent ●●○○○ 20 ptsA standard 52-card deck is shuffled uniformly at random. Find the probability that the ace of spades and the ace of clubs are next to each other in the deck, in either order. Round to four decimal places.
8. Athletes taking someone else's ball ●●○○○ 20 ptsFive athletes came to training, each with their own ball. When leaving, each athlete took a ball that was not their own, and each ball was taken by exactly one athlete. In how many ways is this possible?
9. Birthday match among three people ●●○○○ 20 ptsThree people are in a room. Assuming each birthday is equally likely among 365 days (ignore leap years), find the probability that at least two of them share the same birthday. Round your answer to 9 decimal places.
10. Smallest group with a shared birthday ●●●○○ 50 ptsWhat is the smallest number of people in a group such that the probability that at least two of them share the same birthday is greater than 50%? Assume there are 365 equally likely birthdays and ignore leap years.

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