Final answer:
In 'Gifted Hands,' the affiliation that helped Ben Carson in Australia was with the Seventh-day Adventist Church, which provided him with a vital professional network.
Step-by-step explanation:
In Gifted Hands: The Ben Carson Story, Ben Carson and his wife Candy's affiliation with the Seventh-day Adventist Church was a significant factor that opened doors for them in Australia. Carson's membership in the church connected him with other Adventists, including a renowned Australian surgeon who extended an invitation for Carson to work at a hospital in Perth. This affiliation plays a critical role in the narrative, showcasing the importance of personal networks and shared beliefs in professional opportunities.