Writing N-Letter Words with A, B, and C | Avoiding Repeated A's in Combinations

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Homework Help Overview

The discussion revolves around the problem of counting n-letter words formed from the letters A, B, and C, with the constraint that no two A's can be adjacent. Participants explore various mathematical approaches to tackle this combinatorial problem.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • The original poster considers the potential use of generating functions or alternative methods to solve the problem. One participant introduces a difference equation to express the number of valid words, while others seek clarification on the initial equations presented.

Discussion Status

The discussion includes attempts to clarify the mathematical reasoning behind the equations proposed. Some participants express uncertainty about the clarity of the initial equations, while others assert their validity based on counting possibilities. The conversation appears to be ongoing, with insights being shared.

Contextual Notes

Participants are working within the constraints of a homework problem, which may limit the information they can provide or the methods they can use. There is a focus on ensuring that the solutions adhere to the specified conditions regarding the placement of A's.

heman
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How many n-letter words can be written with A,B,C without placing two A's
adjacent to each other?

will this question involve some kind of genereating function...if not,,how to do it?
 
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You can do it as a difference equation.

Let w(n) be the number words with no repeated A's of length n.

any such can start with a B or C and be followed by any other word of n-1 letters with no repeated As, thus

w(n)=2w(n-1)+no. of words starting with A.

how many words start with A? Well, the next letter can be either B or C, then there is any of the words of length n-2 with no repeated As, thus

w(n)=2w(n-1)+2w(n-2)

subject to the initial conditions w(1)=1, w(2)=8 (3 choices of first letter, 3 of second, minus the one double A choice).
 
just bit more insight ...the first equation is not clear,,
 
just bit more insight ...the first equation is not clear,,
 
yes it is. think about it. I've just 'counted the possibilities'
 
its done..Thanks Matt.
 

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