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1. Two-digit primes from given digits ●○○○○ 10 ptsHow many two-digit prime numbers can be formed by choosing two different digits from the set $\{2, 7, 8, 9\}$ to be used as the tens digit and units digit?
2. 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 $n$.
3. Power of a prime with prime shift ●●○○○ 20 ptsLet $p$ be a prime number such that $p + 3$ is also prime. Find $p^{11} - 52$.
4. Prime sum of white and red balls ●●○○○ 20 ptsA bag contains 6 red balls and 8 white balls. Five balls are taken at random and placed in box $A$, and the remaining 9 balls are placed in box $B$. Find the probability that the sum of the number of white balls in box $A$ and the number of red balls in box $B$ is not a prime number. Round to four decimal places.
5. Primes from up to two digits ●●○○○ 20 ptsPick one or two different digits from the set $\{1, 3, 6, 7\}$ and arrange them to form a number. How many prime numbers can be created in this manner?
6. Distinct values from parenthesised divisions ●●●○○ 50 ptsHow many different values can be obtained by placing parentheses in all possible ways in the expression $2:3:5:7:11:13:17:19:23:29$, where $:$ denotes division?