Question 89
Let $k$ be a non-negative integer and let $A$ be a nonempty finite set. We define a word from the alphabet $A$ of length $k$ as a sequence of elements of $A$ of length $k$. (Notice that by definition the empty sequence is a word of length $0$.)
* Let $B$ be the set of all words from the nine-element alphabet $\{b, @, l, t, i, c, w, a, y\}$, whose lengths are at most $2023$.
* Moreover, let $M$ be the set of all words from the four-element alphabet $\{m, a, t, h\}$, whose lengths are at most $2025$ and which contain the element $t$ exactly twice.
(For example, $tata \in M$, because it has length $4$ and contains the letter $t$ twice, but $mahatma \notin M$, because it contains the letter $t$ only once.)
Prove that the difference $|B| - |M|$ is divisible by $3^{2023}$.
* Let $B$ be the set of all words from the nine-element alphabet $\{b, @, l, t, i, c, w, a, y\}$, whose lengths are at most $2023$.
* Moreover, let $M$ be the set of all words from the four-element alphabet $\{m, a, t, h\}$, whose lengths are at most $2025$ and which contain the element $t$ exactly twice.
(For example, $tata \in M$, because it has length $4$ and contains the letter $t$ twice, but $mahatma \notin M$, because it contains the letter $t$ only once.)
Prove that the difference $|B| - |M|$ is divisible by $3^{2023}$.