The question asks to identify the correct pair of auxiliary verbs for a conditional sentence structure implied by the given clauses.

Identifying Conditional Sentence Structure

The sentence fragment discusses hypothetical situations. The phrase "I wouldn't have had any problem" strongly suggests a past unreal conditional (third conditional) or a related structure. We need to find the correct auxiliary pair to complete such a structure.

Analyzing Auxiliary Verb Pairs

The options provide pairs of auxiliary verbs. The correct answer is specified as 'Had, would'. Let's analyze how this pair functions in conditional sentences.

Matching Auxiliaries to Inverted Mixed Conditional

The pair $Had, would$ perfectly fits the structure of an inverted mixed conditional sentence.

• The structure is: $Had + Subject + Past Participle$, followed by $Subject + would + Base Verb$.
• This structure connects a hypothetical condition in the past (using Had) with a result in the present or future (using would).

For example: "Had I studied diligently, I would have a better job now." (Note: The example here slightly deviates to illustrate the typical 'would' usage in the result clause of a mixed conditional). The pair 'Had, would' signifies this specific grammatical construction.

Conclusion on Auxiliaries

Based on standard English grammar rules for conditional sentences, specifically the inverted mixed conditional, the auxiliary pair $Had, would$ is the correct choice for constructing hypothetical past conditions with present/future results.