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> What is the number of $3$-Letter words from the alphabet {A, B, C} that have no $2$ "A's" directly one after the other?
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What am I doing wrong? I have the following calculation: 

$(1 \cdot 2 \cdot 3) + (2 \cdot 1 \cdot 2) + (3 \cdot 2 \cdot 1) = 16$. 

In each of the above parenthesis, $1$ stands for "A", and since no two "A's" are allowed one after the other, we multiply $1$ with the only $2$ options left (B and C). $3$ is a free slot in which we can use any letter from our alphabet. 

This calculation is wrong, but I'm not sure why. My textbook gives $22$ as a solution. Where is my mistake?