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 == | ||
+ | |||
+ | 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 == |
Revision as of 13:53, 15 March 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 |
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