Question

Define a good word as a square of letters that consists only of the letters A, B and C and in which A is never immediately followed by B, B is never immediately followed by C, and C is never immediately followed by A. If the number of letter good words is 384, find the value of n.

Answer 1

Hint: In this problem, first we need to find the possibility of the letters after A, B and C. Next, write 384 in factored form. Then, compare the factored form of 384 with \[n\] letter good words.

Complete step-by-step answer:
Total number of letter in good word is three, i.e. A, B and C.
The letter A is never immediately followed by B, therefore, there are possibilities of two letters i.e. either A or C.
The letter B is never immediately followed by C, therefore, there are possibilities of two letters i.e. either B or A.
The letter C is never immediately followed by A, therefore, there are possibilities of two letters i.e. either C or B.
The numbers of good letter words are 384.
Now, 384 can be written in factorize form as shown below.
\[384 = 3 \times {2^7}\]
Now, the total number of possibilities of letters along with A, B, and C is 2. Therefore,
\[{\text{The number of n letter good words}} = 3 \times {2^n}\]
Now, compare \[3 \times {2^n}\] with \[3 \times {2^7}\] to obtain the value of\[n\].
\[n = 7\]
Thus, the value of \[n\] is 7.

Note: The factorization of 384 can be obtained using L.C.M. method also. The number of \[n\] letters with good words will be \[3 \times {2^n}\].