[solved]

[Starry]

DNA is formed from bases: A, T, G, C. The order in which the letters are arranged is important, but because a molecule can move, there is no difference between a sequence and the same sequence reversed. For example, the sequence (A, A, T, A, G, A, T) is the same as the sequence (T, A, G, A, T, A, A). How many distinct DNA sequences of 9 bases are there?




I tried stuff like 4^9/2... which isn't right.

 

[Deleted member 4993]

Re: Statistics Help

DNA is formed from bases: A, T, G, C. The order in which the letters are arranged is important, but because a molecule can move, there is no difference between a sequence and the same sequence reversed. For example, the sequence (A, A, T, A, G, A, T) is the same as the sequence (T, A, G, A, T, A, A). How many distinct DNA sequences of 9 bases are there?




I tried stuff like 4^9/2... which isn't right.

Can you have a sequence like (A,A,A,A,A,A,A,A,A) Or (B,A,A,A,A,A,A,A,B) Or (B,B,A,A,A,A,A,B,B) - ETC???

 

[Loren]

Re: Statistics Help

I think your first step is to determine if this is a permutation, a combination or something else. If it is a permutation or combination, simply apply the correct formula.

 

[DrMike]

Burnside's counting lemma (also known as "The Lemma That Is Not Burnside's" - see http://en.wikipedia.org/wiki/Burnside's_lemma) will help here.

Your 'group' is the two operations : I="Do nothing" and R="reverse the sequence"

Your set X is 'All Sequences of 9 bases'

Your sets of fixed points are X[sup:2fkudgfc]I[/sup:2fkudgfc] = X, and X[sup:2fkudgfc]R[/sup:2fkudgfc] = all sequences that read the same backwards and forwards.

The sizes of these are not hard to work out. Once you have, the counting lemma says that your answer is

\(\displaystyle \frac{1}{2}\left[|X^I|+|X^G|\right]\)

 

[galactus]

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