In this question, you are being asked to identify under whose administration the Department of Atomic Energy falls.

Option 1 suggests it is under the Prime Minister`s Office. This is actually the correct answer. The Department of Atomic Energy (DAE) in India falls under the direct control and administration of the Prime Minister`s Office (PMO).

Option 2, the Cabinet Secretariat, is incorrect as its primary role is to provide secretarial assistance to the Cabinet and its committees, and not to administer specific departments.

Option 3 is the Ministry of Power. This is also incorrect as the Ministry of Power primarily handles all policy matters related to power generation and distribution but does not oversee the DAE.

Option 4 is the Ministry of Science and Technology. While it may seem like an apt choice due to the scientific nature of atomic energy, it is incorrect. The Ministry of Science and Technology primarily promotes research and development in the field of science and technology, but the administration of the DAE falls under the jurisdiction of the PMO.

So, the correct answer is the Prime Minister`s Office (1).