2003 AIME II Problems/Problem 3

Problem

Define a $good~word$ as a sequence of letters that consists only of the letters $A$ , $B$ , and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$ , $B$ is never immediately followed by $C$ , and $C$ is never immediately followed by $A$ . How many seven-letter good words are there?

== Solution ==The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

There are three letters to make the first letter in the sequence. However, after the first letter (whatever it is), only two letters can follow it, since one of the letters is restricted. Therefore, the number of seven-letter good words is $3*2^6=192$

Therefore, there are $\boxed{192}$ seven-letter good words.

2003 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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2003 AIME II Problems/Problem 3

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