Difference between revisions of "2003 AIME II Problems/Problem 3"

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Therefore, there are <math>\boxed{192}</math> seven-letter good words.
 
Therefore, there are <math>\boxed{192}</math> seven-letter good words.
 
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== See also ==
 
== See also ==
 
{{AIME box|year=2003|n=II|num-b=2|num-a=4}}
 
{{AIME box|year=2003|n=II|num-b=2|num-a=4}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:54, 13 March 2015

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

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.

See also

2003 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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