The correct answer is: Rajnagar.
Vikramshila University was a Buddhist university in ancient India. It was founded in the 8th century by King Dharmapala of the Pala Empire. The university was located in the town of Bihar Sharif, which is now in the Indian state of Bihar.
Vikramshila University was one of the most important centers of learning in ancient India. It attracted students from all over the world, and its faculty included some of the most renowned scholars of the time. The university was known for its studies in Buddhist philosophy, logic, and Sanskrit literature.
Vikramshila University was destroyed by the Muslim invaders in the 12th century. However, its legacy continues to this day. The university is remembered as one of the greatest centers of learning in the history of India.
Dumraon, Rajnagar, Gidhaur, and Palamu are all princely states in India. However, there is no evidence that any of these states were home to a branch of Vikramshila University.