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

  • Thread starter heman
  • Start date
In summary, to find the number of n-letter words that can be written with A, B, and C without placing two A's adjacent to each other, you can use a difference equation. The equation is w(n) = 2w(n-1) + 2w(n-2), with the initial conditions w(1) = 1 and w(2) = 8. This equation counts all the possibilities of words starting with B or C, followed by any other word of length n-1 with no repeated A's, and also takes into account the number of words starting with A.
  • #1
heman
361
0
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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  • #2
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).
 
  • #3
just bit more insight ...the first equation is not clear,,
 
  • #4
just bit more insight ...the first equation is not clear,,
 
  • #5
yes it is. think about it. I've just 'counted the possibilities'
 
  • #6
its done..Thanks Matt.
 

What are "N letter words"?

"N letter words" are words that contain a specific number of letters, where the number of letters is represented by the letter "N". For example, "cat" is a 3 letter word, while "elephant" is an 8 letter word.

How many "N letter words" are there?

The exact number of "N letter words" depends on the specific value of "N". However, if we consider all possible values of "N", then there are an infinite number of "N letter words".

What are some examples of "N letter words"?

Some examples of "N letter words" include "dog" (3 letters), "computer" (8 letters), and "banana" (6 letters).

Why are "N letter words" important?

"N letter words" are important because they are the building blocks of language. By understanding and using "N letter words", we can effectively communicate with others and express our thoughts and ideas.

How do "N letter words" impact language development?

Learning and using "N letter words" is an essential part of language development. As we grow and learn to read and write, our vocabulary expands with the addition of more complex "N letter words". This allows us to express ourselves more accurately and effectively.

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