Step 1: Understand fatty acids.
A fatty acid is a long carbon chain with a carboxyl group ($-COOH$) at one end. Each fatty acid has a fixed number of carbon atoms.
Step 2: Recall arachidonic acid.
Arachidonic acid is a 20-carbon fatty acid. So its total carbon count is 20.
Step 3: Recall palmitic acid.
Palmitic acid is a 16-carbon fatty acid. So its total carbon count is 16.
Step 4: Read the question coding.
The question pairs each acid with a coded letter standing for its carbon number. Here the code B stands for 20 carbons and A stands for 16 carbons, matching arachidonic acid and palmitic acid in that order.
Step 5: Match to the option.
So arachidonic acid maps to B and palmitic acid maps to A. The option that gives this pair is B, A.
Step 6: Conclude.
Following the carbon counts and the given coding, the correct pairing is B, A.
\[ \boxed{\text{B, A}} \]