BBAAABA and AAAAAAA are examples of "words" that are seven letters long where each letter is either A or B. How many seven-letter words in which each letter is either A or B do not contain three or more A's in a row?
Since each word is to be 7 letters long and there are two choices for each letter, there are $2^{7}=128$ such words.

We count the number of words that do contain three or more A's in a row and subtract this total from 128 .

There is 1 word with exactly 7 A's in a row: AAAAAAA.

There are 2 words with exactly 6 A's in a row: AAAAAAB and BAAAAAA.

Consider the words with exactly $5 \mathrm{~A}$ 's in a row.

If such a word begins with exactly $5 \mathrm{~A}$ 's, then the 6th letter is B and so the word has the form AAAAAB $x$ where $x$ is either $\mathrm{A}$ or $\mathrm{B}$. There are 2 such words.

If such a word has the string of exactly 5 A's beginning in the second position, then it must be BAAAAAB since there cannot be an A either immediately before or immediately after the 5 A's.

If such a word ends with exactly 5 A's, then the 2 nd letter is B and so the word has the form $x$ BAAAAA where $x$ is either $\mathrm{A}$ or $\mathrm{B}$. There are 2 such words.

There are 5 words wth exactly 5 A's in a row.

Consider the words with exactly 4 A's in a row.

Using similar reasoning, such a word can be of one of the following forms: AAAAB $x x$, BAAAAB $x$, $x$ BAAAAB, $x x$ BAAAA.

Since there are two choices for each $x$, then there are $4+2+2+4=12$ words.

Consider the words with exactly 3 A's in a row.

Using similar reasoning, such a word can be of one of the following forms: AAAB $x x x$, BAAAB $x x$, $x \operatorname{BAAAB} x, x x$ BAAAB, $x x x$ BAAA.

Since there are two choices for each $x$, then there appear to be $8+4+4+4+8=28$ such words. However, the word AAABAAA is counted twice. (This the only word counted twice in any of these cases.) Therefore, there are 27 such words.

In total, there are $1+2+5+12+27=47$ words that contain three or more A's in a row, and so there are $128-47=81$ words that do not contain three or more A's in a row.

ANSWER: 81