# Number of 'good words' that can be formed?

A “good” word is any seven letter word consisting of letters from {A,B,C} (some letters may be absent and some letter can be present more than once), with the restriction that A cannot be followed by B, B cannot be followed by C, and C cannot be followed by A. How many good words are there?

This question has already been answered here: How many good words are there?

BUT i have doubts regarding the solution:

You can choose the first letter in 3 ways and any of the other letters in 2 ways, so there can be 3∗$2^6$=192 good words

The remaining six letters can be chosen in 6 ways each? But if we choose A at first place , then we can have A's following it , but once we put a 'C' in some position, it can't be followed by any more A's because "C cannot be followed by A"!

So, once we stumble upon a 'C' , we have only 1 possible letter for the remaining positions- which is "C".

I hope i made myself clear, any help will be highly appreciated.