Patna College was established in 1863 — option 2.

Its place in Bihar’s history. It is the oldest college in Bihar and among the oldest in eastern India, and for a long time it was the principal institution of higher education in the province. It grew out of the Patna High English School, and was affiliated first to Calcutta University; when Patna University was established in 1917 — the seventh university in British India — Patna College became its most important constituent college.

Its alumni. The college educated much of the leadership of Bihar in the national movement and afterwards — among them Rajendra Prasad, the first President of India, and Jayaprakash Narayan. That is why the institution matters in the history of the province and not merely in the history of education.

The wider context. Higher education in British India began with the founding of the universities of Calcutta, Bombay and Madras in 1857, on the recommendation of Wood’s Despatch of 1854, which is called the Magna Carta of English education in India. Patna College was founded within a few years of that reorganisation, as the demand for English-educated clerks, lawyers and officials spread up the Ganga valley.

A note on the distractors. All three years offered are close together and all are plausible for a mid-nineteenth-century foundation; there is no way to reason to the answer, and 1863 has simply to be remembered as one of the standard dates of Bihar general knowledge.

Hence, the answer is 1863.