Hey Rahul

I would like to suggest you that you should go for NIT Trichy rather than that of the IIT Dhanbad.

NIT Trichy is quite known for CS branch and its placements are mesmerising. It will provide you a healthy learning environment. But if you want to look for IIT , its campus and what does it actually feels to be an IITian then you may go for it as from the past few years it has actually improved a lot for it's placement. 

Good luck.

Obviously you should go for IIT Dhanbad Mathematics and Computing over NIT Trichy Computer Science and Engineering.

Especially if you are from north then you would be much more comfortable at IIT Dhanbad. The IIT tag is far bigger than NIT tag and IIT Dhanbad has improved tremendously after getting the IIT conversion and even before it was a well established institute. 

These days Mathematics and computing students are getting better placement oppurtunities as compared to plain computer science graduates.

All the best for your decision!