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Having worked both in public sector and private sector banks, I will try to compare both exam below: -
To tell you frankly, pressure in public sector bank is more or less equivalent to private sector at officer/Manager level. At clerical level, I can say that pressure will be less and they work till the official office hours, but growth wise it will take lot of time to reach managerial level. Following is the typical career path in public sector and private sector banks:
Clerk Officer Manager; Senior Manager; Chief Manager; AGM ;DGM ;GM ;ED ;CMD
When you join ICICI PO, you will be at Assistant Manager level i.e., 2 levels above officer level because in ICICI there is no clerical grade. In terms of promotion, public sector banks emphasize on experience i.e., every three years you will be eligible for a promotion. In private sector, the promotion policy keeps changing based on Bank's outlook of the industry and market needs. If Bank feels that they need people with a specific set of skills, then promotions will be accelerated. But the culture in private sector is more open. The chances of you getting into sales are 50:50 as ICICI has many groups which does not deal with customer facing. Transfers were regular in PSU Banks whereas based on need and available openings, people move across locations in private sector (it is not compulsory). In private sector, if your performance is one of the best then they will provide opportunities to move across verticals. For women, IBPS Clerk is best exam for preparation and getting a job. Monetary wise, private sector is always better and if job safety is preferable for you then public sector is better. Last but not least, those who join PSU Banks recently are not eligible for pension. Hope I clarified broadly on the different choices.

If you are looking for best University for to pursue higher education, you should not be area or location specific, I would advise you to choose best University on the basis of University ranking, placements, infrastructure, faculty members etc. All colleges showcase their achievements but you need to dig deep and choose as if you will get into good college then it will be easy for you to get good skills and get placed into renowned companies.
MBA is considered one of the prestigious programs with a vast majority of students enrolling for different courses under it. There are several specializations in MBA like MBA in Banking, MBA Healthcare Management, MBA in Airport Management, MBA in Information Technology, MBA in Entrepreneurship, etc. MBA also has a great scope and future career opportunities in the field of human resources.
Below are some of the best Private Universities:-
BITS Pilani,
ICT Mumbai, Mumbai.
VIT Vellore, Vellore.
Thapar Institute of Engineering and Technology - [TIET], Patiala.
ITER Bhubaneswar, Bhubaneswar.
LDCE Ahmedabad - LD College of Engineering.
Geeta University, Panipat
MSRIT Bangalore - Ramaiah Institute of Technology.
BMSCE Bangalore - BMS College of Engineering.