Electrical and Electronics Engineering

Yes, you can. But, the criteria usually varies from university to university. So, visit official website or call the admission counselor in which you are seeking admission to have detailed information regarding the same. The top colleges are:
Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore
International Institute of Aerospace Engineering & Management (IIAEM-Jain University), Bangalore
Indian Institute of Science (IISc), Bangalore
Lovely Professional University, (LPU) Punjab
Indian Institute of Space Science and Technology, Thiruvananthapuram IIST Thiruvananthapuram
Indian Institute of Engineering Science and Technology, Shibpur (IIEST-S), Howrah
Madras Institute of Technology, Anna University, Chennai
Birla Institute of Technology, Mesra, Ranchi
Defence Institute of Advanced Technology (DIAT/IAT), Pune.

ECE and EEE are the branches which sound quite similar.
EEE actually goes deeper than just "motors & generators". You also learn more about electromagnetics and how high-frequency signals operate - this is the black-magic that's required to design computer and wireless Systems, one can make more money as an EEE.
ECE: Here, it deals with electronics with some part of communication. The subject uses the scientific knowledge of the behavior and effects of electrons to develop devices, systems, or equipment that uses electricity as part of its driving force. Also, it contains basics on electrical machines and more on integrated circuits and communication systems. As far as scope is concerned, both EEE and ECE are evergreen branches. ECE students can get into IT sectors easily, while the demand of EEE students is high in core companies.
Personally, I feel that the stream might be unique but the only thing a student should focus on is the future goal. Once, he is sure with that, he can then choose the stream of his choice. So, I suggest ECE branch rather than EEE and it also depends upon your choice whether to choose ECE branch and shift to either (software, core) or else go deeper with the subject and choose EEE branch and target a job in the core companies.