I have secured a rank of 277861 in jee mains and I've filled most of the choices please say is there any chance of getting a government college or gov-funded clg. Thank you in advance. Hope for you positive response.

Dear parent,

First you must mention your category that grin which category you have secured this much rank.

Assuming that you have secured 277861 CRL rank in JEE mains 2020 from general category, then with this much rank in JEE mains the chances are negligible for you to get seat in NITs or any iiits in whole josaa round counseling.

Even with your home state quota, there ie no chance taht you can get goverment colleges in josaa counselling not even nit durgapur.

But there is some low chance that you can get in some gftis like mizoram university, gurkula kangri vishwavidyalaya, Ghani Khan Choudhury Institute of Engineering & Technology, etc. with this much rank in csab special round counselling.
At csab special round counseling also there is no chance that you can get nits or iiits or any good gfti like bit mesra.
Also In csab special round there is negligible chances that you can gftis like Sant Longowal Institute of Engineering and Technology, etc if you belong from general category.

And also check out the following link to know more about csab special round counselling and the previous year cutoff Trends ( note that all the mentioned ranks are CRL rank not category wise ranks ) :-
https://engineering.careers360.com/articles/csab-special-round-for-nits-iiits-and-gftis

Also check out our jee mains college predictor tool to get an appropriate idea of the colleges that your son can get on basis of his jee mains rank :-
https://engineering.careers360.com/jee-main-college-predictor?icn=QnA&ici=qna_answer