Question

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 amounts of illumination is
(a) 12² − 1
(b) 2¹²
(c) 2¹² − 1
(d) none of these.

Answer 1

(c) 2¹² − 1
Each of the bulb has its own switch, i.e each bulb will have two outcomes − it will either glow or not glow.
Thus, each of the 12 bulbs will have 2 outcomes.
∴ Total number of ways to illuminate the room = 2¹²

Here, we have also considered the way in which all the bulbs are switched-off. However, this is not required as we need to find out only the number of ways of illuminating the room.
Hence, we subtract that one way from the total number of ways.
= 2¹² − 1