An alien language has $n$ letters $b_1, \dots , b_n$. For some $k < n/2$ assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. Such a $k$-word is considered sacred if:
no letter appears twice and,
if a letter $b_i$ appears in the word then the letters $b_{i−1}$ and $b_{i+1} $ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$.)
For example, if $n = 7$ and $k = 3$ then $b_1b_3b_6, b_3b_1b_6, b_2b_4b_6$ are sacred $3$-words. On the other hand $b_1b_7b_4, b_2b_2b_6$ are not sacred. What is the total number of sacred $k$-words?
Use your formula to find the answer for $n = 10$ and $k = 4$.
I'm stuck at this question for over 3 days. I couldn't make the function so I tried doing it with cases but I still couldn't get the correct answer.
The answer is $600$. Can somebody please explain to me how to do it? And in which category of combinatorics do these fall in?