If by direct admission you mean not having to go through JEE Mains and the subsequent counseling, getting admission based on your board result, and a direct application to BIT Mesra, you can go for its 5 year integrated BSc.+MSc. program.
This course is offered by BIT Mesra in these subjects- Mathematics and Computing, Physics, Chemistry and Food Technology. You will have an exit option after the completion of your 3rd year. The integrated BSc.+MSc. course in Physics/ Chemistry from BIT Mesra is suitable for a career in research, while the integrated BSc.+ MSc. course in Mathematics and Computing/Food Technology offers good placements. You can also opt for research after your degree in Mathematics and Computing or Food Technology.
Thus, for a direct admission into BIT Mesra, you can go for their integrated BSc.+MSc. course.