You are asking about general tickets.
If so, yes you need not tell which train but you have to inform about Passenger train/Express train/ Super fast train.
You can travel in any train in which the class of seats are available for your ticket with out any reservation. You can also travel in reservation coach by paying the difference if reserved accommodation is available and contact TTE before boarding the reserved coach. The train you board should not have any distance restriction (more than your ticket distance.)