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When 'Aa' combine with 'Aa', the number of possible combinations is 3

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When 'AaBb' combines with 'AaBb' the number of possible combinations is:

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I hope I didn't make any mistakes, but you get the idea.

How many combinations if I have 15 characters? 'AaBbCcDdEe...'. The answer is $3^{15}$ but why?

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You only need to understand the first step: AA, Aa and aa are the only 3 combinations possible for one locus. Wether you call it B or C doesn't matter. Now, if you make a long chain of such, you just multiply all the possibilities: $3 \times 3 \times \ldots 3$. So if you have 15, then you should have $3^{15}$. –  Raskolnikov Feb 17 '12 at 14:35
    
In the example given, Aa and aA are obviously assumed to be the same. However, a sequence of genes (or simply switches that can hve two states) would result in the following: AA, Aa, aA and aa –  Pete Feb 2 '13 at 18:39
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up vote 3 down vote accepted

Terminology: I'll refer to {AA, Aa, aa} as the character pairs for the letter A (and similarly for other letters).

The possibilities for each character pair in the combinations of $n$ character pairs are independent. So the total number of combinations is the product of the the number of combinations for each character pair. In this case, they are all the same, since which letter you use does not affect how many pairs you can get.

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