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First of all, none of the institute can offer you integrated MBA for the duration of 4 years. After AICTE directive, the minimum duration of any integrated MBA program is minimum 5 years whether you do a BBA+MBA or B.Tech+ MBA
Following are the options:
Nirma University (tier 3 institute)
NMIMS Mumbai offered in family business (no placement as this meant for those who already have family run business).
MBA Tech in IT Electronics and telecom, Chemical, Computer engineering, manufacturing and mechanical engineering, Civil engineering (5 years), open only for those with 10+2 with physics chemistry and mathematics.
B.Pharma + MBA pharma tech only for those with 10+2 with physics, chemistry and biology.
University of Goa (tier 3)

Join Chandigarh University and gift your learning and career the best institute. University School of Business (USB), ranked among the Best Management Colleges in Punjab has earned recognition for its unique style of teaching & experiential learning. The attribute that makes learning business management exclusive at USB is its specialized approach paired with a research focus, innovation culture and entrepreneurship zeal backed by world-class facilities. Recognizing USB's high standing among the best B-School in Punjab and India, various top-notch companies have collaborated with us to have the best industry-sponsored courses and lab facilities. India's evolution as an emerging economic powerhouse is proving to be fertile ground for MBA graduates because it is easier to navigate unstructured markets with a structured mind. Companies are constantly looking to expand in India and other such emerging economies, and they need professionals trained in business and management to give direction to their growth. Chandigarh University upgrades your professional portfolio with industry-aligned programs, dynamic fraternity, state-of-the-art infrastructure facilities, impeccable placements and much more. -Skilled managers are in demand in an array of industries. Managers with the appropriate experience and credentials are some of the world's the highest-paid professionals. -Adding managerial responsibilities to your job function will result in increased compensation. However, many firms also offer compensation growth options for their valued individual contributors, so moving to management is not the only way to grow your income. -Employment of management occupations is projected to grow 7 percent over the next ten years, faster than the average for all occupations, which will result in about 706,900 new jobs.

IBS Dehradun maintains a reputation for quality teaching, field based assignments, case-based learning, productive learning based internships and final placements in reputed organizations, so you should opt for integrated MBA in IBS ( ICFAI - Dehradun). The Career Management Center at IBS guides and mentors the students on placement matters. IBS Dehradun has achieved excellent campus placement for the final year students and summer internship placements, reflecting the faith and confidence of the industry in the students. Average CTC (in Lacs per annum) for the class of 2021 was 5.62 and the highest CTC (in Lacs per annum) for the class of 2021 was 10.62. Prominent Recruiters are ICICI Bank Ltd. , Axis Bank Ltd. , Berger Paints, Tata Consultancy Services, Byju's, ICICI Prudential Life Insurance, Home First Finance Company and many more.