Select the most appropriate option to fill in the blank. If no article is…

Concept: English uses the indefinite article (‘a’/‘an’) before a singular countable noun that is not yet specific or already known, the definite article (‘the’) before a noun that refers to one specific, already-identified referent, and no article before an uncountable or general-plural noun. Between ‘a’ and ‘an’, the choice depends on the SOUND (not the spelling) of the word that immediately follows: ‘a’ before a consonant sound, ‘an’ before a vowel sound.

Application: In ‘He’s looking for ______ solution to the problem,’ ‘solution’ is a singular countable noun, and nothing earlier in the sentence identifies one particular, already-known solution — the sentence describes an open search for any workable answer. That calls for the indefinite article. Since ‘solution’ begins with the consonant sound /s/, the indefinite article that fits is ‘a’, giving ‘He’s looking for a solution to the problem.’

Cross-check: Testing each alternative against the sentence confirms this:

• ‘No article required’ would need ‘solution’ to be uncountable or used as a general plural (e.g., ‘solutions are welcome’); here it is a singular countable noun that needs a determiner.

• ‘the’ would need the sentence to already have identified one specific solution (e.g., ‘the solution we discussed’); nothing in this sentence does that.

• ‘an’ would need the following word to begin with a vowel sound (e.g., ‘an idea’); ‘solution’ begins with the consonant sound /s/.

So ‘a’ is the only option that is both grammatically required (a determiner before a singular countable noun) and phonetically correct (a consonant-sound match): ‘He’s looking for a solution to the problem.’