Verify Truth Table: Switches and LED
Going by the problem definition directly:


D and B turn ON: 
B•D
A turns OFF while C turns ON: 
~A•C
A and B turn ON and the rest turn OFF:
This condition is already covered by condition 4.
C and D turn OFF:
~C•~D
all switches turn ON:
This condition is already covered by 1.


So your final function is:
B•D + ~A•C + ~C•~D

3 ands, 2 ors.