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1. Distinct primes in 20 factorial ●○○○○ 10 ptsHow many different primes appear in the prime factorization of $20!$ (20 factorial)? Recall that $n!$ is the product of the integers from 1 to $n$.
2. Arrangements of six letters ●○○○○ 10 ptsHow many permutations can be made from the set $\{A, E, I, O, U, Y\}$ using all six letters?
3. Arranging identical coins ●○○○○ 10 ptsSix identical coins are arranged in a row. Find the number of arrangements in which four are heads and two are tails.
4. Limit of a factorial quotient sequence ●●○○○ 20 ptsFind $\lim_{n\to\infty} a_n$, where $a_n= \frac{(n+1)!\,(n-1)!}{(2n)!}$.
5. Limit of n factorial over n to the n ●●○○○ 20 ptsFind $\lim_{n \to \infty} \frac{n!}{n^n}$.
6. Limit of a ratio of factorial sums ●●○○○ 20 ptsFind $\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$.
7. Threshold of a functional equation ●●●○○ 50 ptsLet $f$ be a function taking the nonnegative integers to the positive integers such that $f(0) = f(1) = 1$ and $$\frac{f(n) f(m + 1)}{f(m)} + \frac{f(n) f(n - m)}{f(n - m - 1)} = f(n + 1)$$ for all nonnegative integers $n$ and $m$ with $n \ge m + 1$. Find the smallest nonnegative integer $n$ such that $f(n) > 10^6$.
8. Limit of xth root of factorial over x ●●●○○ 50 ptsFind the value of $\lim_{n\to\infty} \frac{\sqrt[n]{n!}}{n}$, where $n$ is a positive integer. Round to two decimal places.