The genetic code can be viewed as a sequence of four letters T, A, G, and C.
There were two parts to the question:
(a) How many 6-letter sequences are there? I just said $\binom{4}{1}^6$, or $\binom{4}{1}$ choices for each letter.
b is where I am having trouble.
(b) How many 6-letter sequences are palindromic, (read the same forward as backward)? I originally thought that: because for each of the first three letters you get a designated letter for the last three letters, that is if the first is A, the last is A, if the second is T, the second to last is T, etc… So we only need to concern how many options there are for the first three letters. So I thought there should be $$\binom{4}{1}^3\text{combinations.}$$
Then I thought that we might be over counting though. I can't really explain why I think that. I just wanted to check. What do you think?
Best Answer
Both (a) and (b) are correct. In part (a) you have $4$ choices for each letter. Since we are concerned with a $6$-letter sequence we have $4^6$ sequences by the Multiplication Principle. In part (b) we can view our $6$-letter palindromic sequence as a $3$-letter sequence since the last three letters are determined by the choice of the first $3$-letter sequence. So we have $4^3$ $6$-letter palindromic sequences by the Multiplication Principle.