In a room, there are $12$ bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amount of illumination is

there are two cases for each bulb (say ON and OFF)
there will be light in the room when atleast one bulb is ON 
No of ways of selecting ON or OFF are 2 for each i.e., $2^{12}$ for 12 bulbs
but in the above ways of selection there will be a case in which all the bulbs are OFF which leaves the room without light 
$\therefore$ the  number of ways to light the room with different amounts of illumination is $2^{12}-1$