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How did Bihar get its name from the Sanskrit word vihara?

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The Mahabodhi Temple at Bodh Gaya, Bihar, India, features a tall, intricately carved central spire surrounded by smaller shrines, trees, and gardens. — Bihar Mahabodhi Buddhist temple, built in the 2nd century ce, Bodh Gaya, Bihar, India.

The name of the Indian state of Bihar is derived from the Sanskrit word vihara, which refers to a Buddhist monastery or temple.12

In ancient times, the region that is now Bihar served as a principal seat of imperial power and a central hub for Indian culture and civilization. The prevalence of numerous Buddhist monastic communities throughout the area was so significant that the region became known as Bihar, reflecting the high concentration of these religious centers. Historically, these monasteries, such as the famous center of learning at Nalanda, were essential to the region's identity and intellectual life.23