Difference between revisions of "2003 AIME II Problems/Problem 3"
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Define a <math>good~word</math> as a sequence of letters that consists only of the letters <math>A</math>, <math>B</math>, and <math>C</math> - some of these letters may not appear in the sequence - and in which <math>A</math> is never immediately followed by <math>B</math>, <math>B</math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? | Define a <math>good~word</math> as a sequence of letters that consists only of the letters <math>A</math>, <math>B</math>, and <math>C</math> - some of these letters may not appear in the sequence - and in which <math>A</math> is never immediately followed by <math>B</math>, <math>B</math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? | ||
| − | == Solution == | + | == Solution 1 == |
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 <math>3*2^6=192</math> | 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 <math>3*2^6=192</math> | ||
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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| + | == Solution 2 == | ||
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| + | There are three choices for the first letter and two choices for each subsequent letter, so there are <math>3\cdot2^{n-1}\ n</math>-letter good words. Substitute <math>n=7</math> to find there are <math>3\cdot2^6=\boxed{192}</math> seven-letter good words. ~ aopsav (Credit to AoPS Alcumus) | ||
== 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}} | ||
| + | |||
| + | [[Category: Intermediate Combinatorics Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 11:15, 20 August 2020
Contents
Problem
Define a 
as a sequence of letters that consists only of the letters 
, 
, and 
- some of these letters may not appear in the sequence - and in which is never immediately followed by , is never immediately followed by , and is never immediately followed by . How many seven-letter good words are there?
Solution 1
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 
Therefore, there are 
seven-letter good words.
Solution 2
There are three choices for the first letter and two choices for each subsequent letter, so there are 
-letter good words. Substitute 
to find there are 
seven-letter good words. ~ aopsav (Credit to AoPS Alcumus)
See also
| 2003 AIME II (Problems • Answer Key • Resources) | ||
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
