The British gained control of Bihar after their victory in the Battle of:

Buxar
Plassey
Panipat
Talikota

The correct answer is: Buxar.

The Battle of Buxar was fought on 22 October 1764 between the British East India Company and the Nawab of Awadh, Shuja-ud-Daula, and the Mughal emperor Shah Alam II. The battle was a decisive victory for the British, and it led to the British gaining control of Bihar, Bengal, and Orissa.

The Battle of Plassey was fought on 23 June 1757 between the British East India Company and the Nawab of Bengal, Siraj ud-Daulah. The battle was a decisive victory for the British, and it led to the British gaining control of Bengal.

The Battle of Panipat was fought on 5 November 1761 between the Maratha Empire and the Afghan Empire. The battle was a decisive victory for the Afghans, and it led to the decline of the Maratha Empire.

The Battle of Talikota was fought on 16 January 1565 between the Vijayanagara Empire and the Deccan sultanates. The battle was a decisive victory for the Deccan sultanates, and it led to the decline of the Vijayanagara Empire.

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