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586 is a very good score. First of all, Congratulations for that!! For your second question, if you hold a domicile certificate of Bihar, then you have a high chance of getting a government medical college in Bihar. AIQ cutoffs of Bihar are quite high and anything less than 600 won't fetch you a college. I don't want to disappoint you but since I've been through this stage I'm sharing my opinion with you.
For more details, check
http://www.google.com/amp/s/medicine.careers360.com/articles/neet-cutoff-bihar/amp
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Hello,
Based on the closing rank of 2018 you have less chance.
The closing rank was 11250.
There will be slight variatons based on the no of seats and no of applicants etc.
You can check the previous year closing ranks using the link below for refrence,
https://medicine.careers360.com/articles/neet-cutoff-bihar
You can take the help of college predictor also,
https://medicine.careers360.com/neet-college-predictor
All the best.
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