Fahien, a Chinese Buddhist monk, embarked on a mission to India with the purpose of visiting Buddhist institutions and collecting copies of Buddhist manuscripts. This option is the correct answer because Fahien`s main objective was to understand and study the Buddhist teachings and practices in India.

Option 1, learning about the administrative system of the Gupta kings, is not the main focus of Fahien`s mission. Although he might have gained some understanding of the administrative system during his travels, it was not his primary objective.

Option 2, understanding the social position of women during the Gupta period, is also not the main purpose of Fahien`s mission. While he may have observed and noted the social dynamics during his journey, it was not his main focus.

Option 4, getting full knowledge about the condition of peasants during the period of Gupta kings, is not mentioned in Fahien`s mission. His primary goal was centered around Buddhism and its practices, rather than examining the conditions of peasants.

In summary, Fahien`s mission to India was primarily focused on visiting Buddhist institutions and collecting Buddhist manuscripts, making option 3 the correct answer.