There are twelve bulbs in a hall, each one of them can be switched independently. The number of ways in which the hall can be illuminated is

The room can be illuminated even if one bulb is illuminated and for each bulb there are 2 choices of being ON or OFF and there are 12 bulbs.
So, number of ways in bulbs can be lighted $=2^{12}$
But, in the above cases, there is one single case in which all the bulbs are off and the room is not illuminated.
So, number of ways of illuminating the room $=2^{12}-1$