How do we know the direction of induced current in a coil due to a current carrying wire or moving electron,with or without relative motion between the coil and the wire?

@ anmol 
you can use lenzes law and faradey right hand thumb rule to calculate the direction of induced current . 
acc to lenzes law , the direction of induced current will be in such a direction so, that it will oppose the cause of it . 

that means if a magnet is coming towards the coil then the current associated with it will be anticlockwise , and it any magnet is moving away from any coil then the current induced in it will be anticlockwise . 

and faradey right hand thumb rule says that the first finger and thum,b and middle finger are in mutual perpendicular direction , then the first finger is along the direction of megnatic field , the middle finger is along the direction of induced e.m .f and the thumb is in direction of FORCE . 
that depend upon the question actually , so, apply anyone of the both acc to the requirment . 

HOPE IT CLEARS YOUR DOUBT 

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