MathDB

\textrm{alien}

script has

n

letters

b_1,b_2,\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

<span class='latex-bold'>sacred</span>

if:
i. no letter appears twice and, ii. 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

Problem Statement