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According to the information given by you, you may easily get B.A. MS BHMS and BDS in government colleges. With 503 marks (OBC NCL), getting into a government college for the MBBS is a little difficult, but you can expect some colleges by round three. You may expect the following colleges:
(a) For the MBBS course:
1. The Atal Bihar Vajpayee Government Medical College
2. Data Meghe Institute of Medical Sciences
(b) For the Bachelor of Dental Surgery BDS Course:
3. Pandit Bhagwat Dayal Sharma University of Health Sciences
4. SRM Kattankulathur Dental College and Hospital
(c) For Bachelor of Homoeopathic Medicine and Surgery (BHMS):
5. Pratap Chandra Memorial Homoeopathic Hospital and College in Kolkata
(d) For a Bachelor of Ayurvedic Medicine and Surgery:
6. Bundelkhand Government Ayurvedic College and Hospital

Yes, the Ram Krishna College of Ayurveda & Medical Science of Bhopal offers a Bachelor of Ayurvedic Medicine and Surgery (B.A. M. S. ). Admissions to this course in college are done on the basis of NEET scores following the NEET counseling process by the M. P. Ayush Department (for 85% seats) and the Ministry of Ayush (for 15% seats). The eligibility criteria for admission to the course is as follows: (a) Candidates should have passed the 12th or equivalent examination with PCB and English as compulsory subjects. (b) In addition, candidates must have scored at least 50% and qualified for the NEET UG exam. It has an intake of 100 seats for this course.

With a 67.33 percentile in MH-CET, you may get admission to select colleges in Mumbai for data Science and related courses. Here are the colleges where you may be eligible based on the previous year's cut-off: (1) A. P. Shah Institute of Technology in Thane West: Here you may apply for a B.Tech. In Computer Science and Engineering (Data Science) course. (2) Universal College of Engineering in Palghar: You can pursue a B.E. In Data Engineering at this college. (3) A. C. Patil College of Engineering in Navi Mumbai: You can pursue a B.E. In Artificial Intelligence and Data Science from this college. You can also use the MH-CET college predictor https://www.shiksha.com/engineering/mht-cet-college-predictor tool to find more colleges and courses that match your score.

No, you cannot get direct entry in MBBS 2nd year based on B.A. M.S. There is no lateral entry scheme applicable for admission in MBBS. You can apply for MBBS based on 12th result but not on B.A. M.S. Since you have already passed B.A. M.S you can apply for higher studies related to B.A. M.S than going for MBBS. MBBS and B.A. M.S are two completely different courses. MBBS is Bachelor of Medicine and Bachelor of Surgery whereas B.A. M.S is Bachelor of Ayurvedic Medicine and Surgery. After completing B.A.M.S you can apply for courses like:-
MS in Ayurveda
M. S in Ayurvedic Dhanvantari
M. D in Ayurvedic
M. D in Ayurvedic Vachaspati
PG Diploma in Balaroga
PG Diploma in Panchakarma
There are also some Master degree and short term courses in the field of Physiology, Anatomy, Microbiology, Surgery, Toxicology, Pharmacology, Naturopathy, Botany, etc. It will improve the knowledge and skills.