Prajnanananda Institute of Technology and Management

Hi Abhishek,
Prajnanananda Institute of Technology provides BCA (Hons) as a full time course:
https://www.shiksha.com/it-software/course/bca-hons-prajnanananda-institute-of-technology-and-management-park-street-kolkata-177894

For BCA, you should have either Mathematics, Computer Science or Business Mathematics (or a related subject).
For fees details, check out website http://www.pitmkol.in

BCA Course: Eligibility
Candidates who wish to pursue BCA course must fulfil the below mentioned eligibility criteria:
Pass in 10+2 examination with any stream from a recognized board of education.
English as a subject of study.
A minimum aggregate score of 45% at 10+2 level.
Some colleges prefer students from science stream or those who have studied mathematics and computer science at 10+2 level.

Yes, you can. There will not be a problem regarding the course.
Only possible problem: during campus placements, some companies may not hold you eligible for recruitment. But nothing to worry, most companies will consider you for placements.