MBA/PGDM

Yes, you can do PGDM in correspondent from Indian Institute of Business Management and Studies.
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Yes, you can do MBA in hospital management.
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No, you cannot pursue executive MBA with 4 years of industrial experience, because they want 5 years' experience.
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You can check the link to get information about the same http://www.iibms.org/course/distance-online-emba/
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This institution is approved by AICTE.
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Yes, you can apply for distance MBA in finance.
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Yes, you can get admission for MBA.
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Please check this link for better results
https://www.shiksha.com/b-tech/resources/kcet-college-predictor
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Admission procedure:
The admissions will be given as per eligibility and first come, first server basis. If candidate apply more the 120 seats then institute conducted the admission test and the admissions will be done as per merit in the admission test conducted by the institute. Good luck.

No, there is no entrance exam for distance MBA. Good luck.