Final answer:
The correct word to use is 'Whoever' since it serves as the subject of the clause in the sentence.
Step-by-step explanation:
The correct word to use in the sentence "(Whoever, Whomever) draws the best picture will win a prize." is Whoever. This word is appropriate here because it's being used as the subject of a clause. In English, when determining whether to use 'whoever' or 'whomever', it is essential to look at the role the word plays in the clause. If it's the subject of the clause, 'whoever' should be used. Examples include "Who is going to the concert?" where 'who' is the subject, and "She is the person who is best qualified for the job," where 'who' operates as the subject in the clause 'who is best qualified for the job'. Conversely, 'whomever' is used as the object of a verb, preposition, or clause. For instance, "Give the tickets to whomever you choose" uses 'whomever' as the object of the clause.