What government colleges are possible for someone with a 27k rank in JEE Mains?

Congratulations on scoring this much rank in Joint Entrance Examination Mains.

Since you have not mentioned your state of eligibility it would be diificult for me to find out colleges for you that you will be getting from your state quota.

I am giving some choices for you

• National Institute of Technology Hamirpur Chemical Engineering
• Dr. B R Ambedkar National Institute of Technology, Jalandhar Chemical Engineering
• National Institute of Technology Sikkim Computer Science
• National Institute of Technology, Silchar Electrical Engineering
• Indian Institute of Information Technology(IIIT), Sri City, Chittoor District, Andhra Pradesh Electronics & Communication

Visit this link to have an idea of more colleges and respective branches that you will be getting on the basis of your rank

http://tools.collegepravesh.com/college-predictor/jee-main/

National Institute of Technology or NIT are the technical institutes at national level offering admission to candidates on the basis of the entrance exam called JEE Main. Candidates will have to appear in JEE Main and qualify it with a good rank to be eligible for admission at NITs. There are 31 NITs in the country having their respective cutoffs (minimum marks required for admission) which they release after the exam is over. Candidates will have to score marks more than or equivalent to respective NITs cutoff, they want to be a part of.

https://engineering.careers360.com/articles/jee-main-cutoff refer this link for last year college cutoffs.