DNA Nucleotides - Permutations and Combinations

I have been working on this problem for 2 days and keep coming back to square 1: I don't know what to do!

The problem reads as follows:

DNA is made of nucleotides and each nucleotide can contain any one of these nitrogenous bases: A,G,C,T. One of those four bases must be selected three times to form a linear triplet. How many different triplets are possible? Note that all four bases can be selected for each of the three components of the triplet.

I just figured out this morning that this means we have the letters ACGT which can be arranged in either 35 or 256 ways (I'm not sure if order matters in this case). For instance, AAAT, AGTA, AAGG, GGGT, GGGA and so on. However, I got this far by plugging it into a formula online and I want to know how to do it by hand. Additionally, I'm not sure how to go about figuring how many of those possible arrangements are linear triplets?

Any help would be GREATLY appreciated as this assignment is due today and this is the last problem I have left - the only one I can't figure out!

Thank you.

For a triplet, you have 4 choices for each of the 3 positions.

Using the letters A, G, C, T there are four positions in any string and each position can be filled in four ways thus there are \(\displaystyle {4}^{4}={256}\) ways to have a string of four.

I am not quite sure about what a linear triple actually is.
Having said that, if AAAG & GAAA are linear triples and AAGA is not then there are (4)(3)(2) ways to pick them. Four ways to pick the letter to be tripled, three ways to pick the single letter and two ways to arrange the.
But I still have two questions. Is AAAA a possible triple? What about AAGA?

Thanks for your replies! The answer was 4^3, though I'm still not sure why. Thanks again!

karrate7785 said:

The answer was 4^3, though I'm still not sure why.

Click to expand...

No that is not the answer to the question as you framed it.
What is the correct wording of the question?
It must be different from what you posted.
Otherwise that is a wrong answer!

Isn't \(\displaystyle 4^3\) equal to the number of possible codons?

All these excerpts from Wikipedia at http://en.wikipedia.org/wiki/Codon:

"...the code defines a mapping between tri-nucleotide sequences called codons and amino acids; every triplet of nucleotides in a nucleic acid sequence specifies a single amino acid."

"There are 4³ = 64 different codon combinations possible with a triplet codon of three nucleotides. In reality, all 64 codons of the standard genetic code are assigned for either amino acids or stop signals during translation."

The article gives examples like this as codons:

AAA, AGA, ACT, or any arrangement of any of the three nucleotide bases.

I still don't know what a "linear triplet" is either. Is it a codon made of all one letter, like AAA? In that case, wouldn't the answer be 4? Though, it seems like a linear triplet is basically a codon. Then, you would have 4 ways to fill-in each blank of the triplet. You would just use the "counting rule".

These problems that have to do with topics that not everyone is familiar with can really throw our minds through a mobius strip. Don't ask me a math question related to baseball or any other sport.

~Kasie

I worded it exactly as it is in my text and that is the answer that was given by the instructor after the assignment was graded.

Like I said before, I don't understand the problem which is why I came searching for help. Honestly, I still don't understand the question, let alone how to arrive at the answer.