A DNA chain is a string of As, Ts, Gs, and Cs. How many DNA
chains are there of length 12 which have the following properties?
Choose two of (a), (b), or (c) to submit:
(a) Is a palindrome (A palindrome reads the same forwards and
backwards, e.g., TAGGAT.)
To find the number of DNA chains of length 12 that are palindromes, we need to consider the properties of a palindrome.
Since a palindrome reads the same forwards and backwards, the first half of the DNA chain will be the same as the second half but in reverse order. Let's break down the process step by step:
Step 1: Determine the number of choices for the first half of the DNA chain.
Since the first half needs to match the second half in reverse order, we need to choose the characters for the first half carefully. For each position in the first half (total of 6 positions), we can choose any of the four characters: A, T, G, or C. Therefore, there are 4 choices for each position, resulting in 4^6 possible choices for the first half.
Step 2: Determine the number of choices for the second half of the DNA chain.
Since the second half needs to match the first half in reverse order, the choices for the second half are already determined by the choices made for the first half.
Step 3: Calculate the total number of DNA chains.
To find the total number of DNA chains, we multiply the number of choices for the first half by the number of choices for the second half. Since the choices for the second half are determined by the choices made for the first half, the total number of DNA chains is the square of the number of choices for the first half.
Therefore, the total number of DNA chains of length 12 that are palindromes is (4^6)^2 = 4^12 = 16,777,216.