Recent work shows that large language models (LLMs) can answer multiple-choice questions using only the choices, but does this mean that MCQA leaderboard rankings of LLMs are largely influenced by abilities in choices-only settings? To answer this, we use a contrast set that probes if LLMs over-rely on choices-only shortcuts in MCQA. While previous works build contrast sets via expensive human annotations or model-generated data which can be biased, we employ graph mining to extract contrast sets from existing MCQA datasets. We use our method on UnifiedQA, a group of six commonsense reasoning datasets with high choices-only accuracy, to build an 820-question contrast set. After validating our contrast set, we test 12 LLMs, finding that these models do not exhibit reliance on choice-only shortcuts when given both the question and choices. Thus, despite the susceptibility of MCQA to high choices-only accuracy, we argue that LLMs are not obtaining high ranks on MCQA leaderboards solely due to their ability to exploit choices-only shortcuts.