Question 12
Consider 26 letters $A, \dots, Z$. A *string* is a finite sequence consisting of those letters. We say that a string $s$ is *nice* if it contains each of the 26 letters at least once, and each permutation of letters $A, \dots, Z$ occurs in $s$ as a subsequence the same number of times. Prove that:
a. There exists a nice string.
b. Any nice string contains at least 2022 letters.
(Here, a permutation $\pi$ of the 26 letters is a *subsequence* of a string $s = s_1s_2s_3\dots$ if there exist 26 indices $i_1 \lt i_2 \lt \dots \lt i_{26}$ such that $\pi = s_{i_1}s_{i_2}\dots s_{i_{26}}$.) (Václav Rozhoň)
a. There exists a nice string.
b. Any nice string contains at least 2022 letters.
(Here, a permutation $\pi$ of the 26 letters is a *subsequence* of a string $s = s_1s_2s_3\dots$ if there exist 26 indices $i_1 \lt i_2 \lt \dots \lt i_{26}$ such that $\pi = s_{i_1}s_{i_2}\dots s_{i_{26}}$.) (Václav Rozhoň)