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1. Limit of binomial coefficient over quadratic ●○○○○ 10 ptsCalculate $\lim_{n \to \infty} \frac{\binom{n}{2}}{n^2+1}$. Give your answer as a decimal.
2. Choosing two balls from twenty ●○○○○ 10 ptsTwenty balls numbered 1 through 20 are in a bin. In how many ways can 2 balls be selected if the order in which they are drawn does not matter?
3. Equal votes in a village election ●●○○○ 20 ptsA small village has $n$ people. During their yearly elections, every group of three people comes up to a stage and, as a group, casts a single vote for someone in the village to be the new leader. After every possible group of three people has voted, it turns out that every person in the village received exactly the same number of votes. If $10 \leq n \leq 100$, how many possible values of $n$ are there?
4. Same number of heads for two flips ●●○○○ 20 ptsJackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
5. Rolling more 1s than 6s ●●●○○ 50 ptsIf Greg rolls four fair six-sided dice, what is the probability that he rolls more 1s than 6s? Round to two decimal places.
6. Partition of a set with size conditions ●●●○○ 50 ptsLet $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$, $\mathcal{A} \cap \mathcal{B} = \emptyset$, the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$, and the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$. Find $N$.
7. Tournament with two dropouts ●●●○○ 50 ptsIn a tournament, each participant was supposed to play exactly one game with each of the other participants. Two participants dropped out during the tournament, each having played only 4 games. In the end, the total number of games played was 62. How many participants were there in total?
8. Counting nine-term sequences with ratio constraints ●●●○○ 50 ptsThe sequence $\{a_n\}$ has nine terms, $a_1=a_9=1$, and for each $i\in\{1,2,\ldots,8\}$ we have $\frac{a_{i+1}}{a_i}\in\left\{2,1,-\frac{1}{2}\right\}$. How many such sequences are there?
9. Matching head counts with unfair coin ●●●○○ 50 ptsJackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
10. Limit of a central binomial ratio ●●●○○ 50 ptsFind $\lim_{n \to \infty} \frac{(2n)!\,\sqrt{n}}{(n!)^2\,4^n}$. Round to two decimal places.