Final answer:
Old age is the risk factor that places patients and residents at the greatest risk for falls.
Step-by-step explanation:
The risk factor that places patients and residents at the greatest risk for falls is old age. As people age, they may experience a decline in vision, balance, and overall physical strength, making them more prone to falling. Additionally, elderly individuals may have underlying health conditions that further increase their risk of falls, such as osteoporosis, which can lead to fractures and mobility issues.